Find the Inverse f(x) = square root of x/(x-1)

$f (x) = \sqrt{\frac{x}{x - 1}}$

Write $f (x) = \sqrt{\frac{x}{x - 1}}$ as an equation.

$y = \sqrt{\frac{x}{x - 1}}$

Interchange the variables.

$x = \sqrt{\frac{y}{y - 1}}$

Solve for $y$ .

Multiply the equation by $y - 1$ .

$(y - 1) (x) = (\sqrt{\frac{y}{y - 1}}) (y - 1)$

Simplify the left side.

Simplify $(y - 1) (x)$ .

Apply the distributive property.

$y x - 1 x = (\sqrt{\frac{y}{y - 1}}) (y - 1)$

Rewrite $- 1 x$ as $- x$ .

$y x - x = (\sqrt{\frac{y}{y - 1}}) (y - 1)$

Simplify the right side.

Simplify $(\sqrt{\frac{y}{y - 1}}) (y - 1)$ .

Rewrite $\sqrt{\frac{y}{y - 1}}$ as $\frac{\sqrt{y}}{\sqrt{y - 1}}$ .

$y x - x = \frac{\sqrt{y}}{\sqrt{y - 1}} (y - 1)$

Multiply $\frac{\sqrt{y}}{\sqrt{y - 1}}$ by $\frac{\sqrt{y - 1}}{\sqrt{y - 1}}$ .

$y x - x = \frac{\sqrt{y}}{\sqrt{y - 1}} \cdot \frac{\sqrt{y - 1}}{\sqrt{y - 1}} (y - 1)$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{y}}{\sqrt{y - 1}}$ by $\frac{\sqrt{y - 1}}{\sqrt{y - 1}}$ .

$y x - x = \frac{\sqrt{y} \sqrt{y - 1}}{\sqrt{y - 1} \sqrt{y - 1}} (y - 1)$

Raise $\sqrt{y - 1}$ to the power of $1$ .

$y x - x = \frac{\sqrt{y} \sqrt{y - 1}}{{\sqrt{y - 1}}^{1} \sqrt{y - 1}} (y - 1)$

Raise $\sqrt{y - 1}$ to the power of $1$ .

$y x - x = \frac{\sqrt{y} \sqrt{y - 1}}{{\sqrt{y - 1}}^{1} {\sqrt{y - 1}}^{1}} (y - 1)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y x - x = \frac{\sqrt{y} \sqrt{y - 1}}{{\sqrt{y - 1}}^{1 + 1}} (y - 1)$

Add $1$ and $1$ .

$y x - x = \frac{\sqrt{y} \sqrt{y - 1}}{{\sqrt{y - 1}}^{2}} (y - 1)$

Rewrite ${\sqrt{y - 1}}^{2}$ as $y - 1$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y - 1}$ as ${(y - 1)}^{\frac{1}{2}}$ .

$y x - x = \frac{\sqrt{y} \sqrt{y - 1}}{{({(y - 1)}^{\frac{1}{2}})}^{2}} (y - 1)$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y x - x = \frac{\sqrt{y} \sqrt{y - 1}}{{(y - 1)}^{\frac{1}{2} \cdot 2}} (y - 1)$

Combine $\frac{1}{2}$ and $2$ .

$y x - x = \frac{\sqrt{y} \sqrt{y - 1}}{{(y - 1)}^{\frac{2}{2}}} (y - 1)$

Cancel the common factor of $2$ .

Cancel the common factor.

$y x - x = \frac{\sqrt{y} \sqrt{y - 1}}{{(y - 1)}^{\frac{2}{2}}} (y - 1)$

Rewrite the expression.

$y x - x = \frac{\sqrt{y} \sqrt{y - 1}}{{(y - 1)}^{1}} (y - 1)$

Simplify.

$y x - x = \frac{\sqrt{y} \sqrt{y - 1}}{y - 1} (y - 1)$

Combine using the product rule for radicals.

$y x - x = \frac{\sqrt{y (y - 1)}}{y - 1} (y - 1)$

Cancel the common factor of $y - 1$ .

Cancel the common factor.

$y x - x = \frac{\sqrt{y (y - 1)}}{y - 1} (y - 1)$

Rewrite the expression.

$y x - x = \sqrt{y (y - 1)}$

Solve for $y$ .

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{y (y - 1)} = y x - x$

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{y (y - 1)}}^{2} = {(y x - x)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y (y - 1)}$ as ${(y (y - 1))}^{\frac{1}{2}}$ .

${({(y (y - 1))}^{\frac{1}{2}})}^{2} = {(y x - x)}^{2}$

Simplify the left side.

Simplify ${({(y (y - 1))}^{\frac{1}{2}})}^{2}$ .

Simplify by multiplying through.

Multiply the exponents in ${({(y (y - 1))}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(y (y - 1))}^{\frac{1}{2} \cdot 2} = {(y x - x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(y (y - 1))}^{\frac{1}{2} \cdot 2} = {(y x - x)}^{2}$

Rewrite the expression.

${(y (y - 1))}^{1} = {(y x - x)}^{2}$

Apply the distributive property.

${(y \cdot y + y \cdot - 1)}^{1} = {(y x - x)}^{2}$

Simplify the expression.

Multiply $y$ by $y$ .

${(y^{2} + y \cdot - 1)}^{1} = {(y x - x)}^{2}$

Move $- 1$ to the left of $y$ .

${(y^{2} - 1 \cdot y)}^{1} = {(y x - x)}^{2}$

Rewrite $- 1 y$ as $- y$ .

${(y^{2} - y)}^{1} = {(y x - x)}^{2}$

Simplify.

$y^{2} - y = {(y x - x)}^{2}$

Simplify the right side.

Simplify ${(y x - x)}^{2}$ .

Rewrite ${(y x - x)}^{2}$ as $(y x - x) (y x - x)$ .

$y^{2} - y = (y x - x) (y x - x)$

Expand $(y x - x) (y x - x)$ using the FOIL Method.

Apply the distributive property.

$y^{2} - y = y x (y x - x) - x (y x - x)$

Apply the distributive property.

$y^{2} - y = y x (y x) + y x (- x) - x (y x - x)$

Apply the distributive property.

$y^{2} - y = y x (y x) + y x (- x) - x (y x) - x (- x)$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$y^{2} - y = y \cdot y x \cdot x + y x (- x) - x (y x) - x (- x)$

Multiply $y$ by $y$ .

$y^{2} - y = y^{2} x \cdot x + y x (- x) - x (y x) - x (- x)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$y^{2} - y = y^{2} (x \cdot x) + y x (- x) - x (y x) - x (- x)$

Multiply $x$ by $x$ .

$y^{2} - y = y^{2} x^{2} + y x (- x) - x (y x) - x (- x)$

Rewrite using the commutative property of multiplication.

$y^{2} - y = y^{2} x^{2} - y x \cdot x - x (y x) - x (- x)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$y^{2} - y = y^{2} x^{2} - y (x \cdot x) - x (y x) - x (- x)$

Multiply $x$ by $x$ .

$y^{2} - y = y^{2} x^{2} - y x^{2} - x (y x) - x (- x)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$y^{2} - y = y^{2} x^{2} - y x^{2} - (x \cdot x) y - x (- x)$

Multiply $x$ by $x$ .

$y^{2} - y = y^{2} x^{2} - y x^{2} - x^{2} y - x (- x)$

Rewrite using the commutative property of multiplication.

$y^{2} - y = y^{2} x^{2} - y x^{2} - x^{2} y - 1 \cdot - 1 x \cdot x$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$y^{2} - y = y^{2} x^{2} - y x^{2} - x^{2} y - 1 \cdot - 1 (x \cdot x)$

Multiply $x$ by $x$ .

$y^{2} - y = y^{2} x^{2} - y x^{2} - x^{2} y - 1 \cdot - 1 x^{2}$

Multiply $- 1$ by $- 1$ .

$y^{2} - y = y^{2} x^{2} - y x^{2} - x^{2} y + 1 x^{2}$

Multiply $x^{2}$ by $1$ .

$y^{2} - y = y^{2} x^{2} - y x^{2} - x^{2} y + x^{2}$

Subtract $x^{2} y$ from $- y x^{2}$ .

Move $y$ .

$y^{2} - y = y^{2} x^{2} - x^{2} y - x^{2} y + x^{2}$

Subtract $x^{2} y$ from $- x^{2} y$ .

$y^{2} - y = y^{2} x^{2} - 2 x^{2} y + x^{2}$

Solve for $y$ .

Move all terms containing $y$ to the left side of the equation.

Subtract $y^{2} x^{2}$ from both sides of the equation.

$y^{2} - y - y^{2} x^{2} = - 2 x^{2} y + x^{2}$

Add $2 x^{2} y$ to both sides of the equation.

$y^{2} - y - y^{2} x^{2} + 2 x^{2} y = x^{2}$

Subtract $x^{2}$ from both sides of the equation.

$y^{2} - y - y^{2} x^{2} + 2 x^{2} y - x^{2} = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1 - x^{2}$ , $b = - 1 + 2 x^{2}$ , and $c = - x^{2}$ into the quadratic formula and solve for $y$ .

$\frac{- (- 1 + 2 x^{2}) \pm \sqrt{{(- 1 + 2 x^{2})}^{2} - 4 \cdot ((1 - x^{2}) \cdot (- x^{2}))}}{2 (1 - x^{2})}$

Simplify.

Simplify the numerator.

Apply the distributive property.

$y = \frac{1 - (2 x^{2}) \pm \sqrt{{(- 1 + 2 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $- 1$ by $- 1$ .

$y = \frac{1 - (2 x^{2}) \pm \sqrt{{(- 1 + 2 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $2$ by $- 1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{{(- 1 + 2 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Rewrite ${(- 1 + 2 x^{2})}^{2}$ as $(- 1 + 2 x^{2}) (- 1 + 2 x^{2})$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{(- 1 + 2 x^{2}) (- 1 + 2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Expand $(- 1 + 2 x^{2}) (- 1 + 2 x^{2})$ using the FOIL Method.

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{- 1 (- 1 + 2 x^{2}) + 2 x^{2} (- 1 + 2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{- 1 \cdot - 1 - 1 (2 x^{2}) + 2 x^{2} (- 1 + 2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{- 1 \cdot - 1 - 1 (2 x^{2}) + 2 x^{2} \cdot - 1 + 2 x^{2} (2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 1$ by $- 1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 1 (2 x^{2}) + 2 x^{2} \cdot - 1 + 2 x^{2} (2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $2$ by $- 1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} + 2 x^{2} \cdot - 1 + 2 x^{2} (2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $- 1$ by $2$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 2 x^{2} (2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Rewrite using the commutative property of multiplication.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 2 \cdot (2 x^{2} x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 2 \cdot (2 (x^{2} x^{2})) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 2 \cdot (2 x^{2 + 2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Add $2$ and $2$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 2 \cdot (2 x^{4}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $2$ by $2$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 4 x^{4} - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Subtract $2 x^{2}$ from $- 2 x^{2}$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (- 4 \cdot 1 - 4 (- x^{2})) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $- 4$ by $1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (- 4 - 4 (- x^{2})) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $- 1$ by $- 4$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (- 4 + 4 x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (- 4 \cdot - 1 + 4 x^{2} \cdot - 1) x^{2}}}{2 (1 - x^{2})}$

Multiply $- 4$ by $- 1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (4 + 4 x^{2} \cdot - 1) x^{2}}}{2 (1 - x^{2})}$

Multiply $- 1$ by $4$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (4 - 4 x^{2}) x^{2}}}{2 (1 - x^{2})}$

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + 4 x^{2} - 4 x^{2} x^{2}}}{2 (1 - x^{2})}$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + 4 x^{2} - 4 (x^{2} x^{2})}}{2 (1 - x^{2})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + 4 x^{2} - 4 x^{2 + 2}}}{2 (1 - x^{2})}$

Add $2$ and $2$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + 4 x^{2} - 4 x^{4}}}{2 (1 - x^{2})}$

Add $- 4 x^{2}$ and $4 x^{2}$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 + 4 x^{4} + 0 - 4 x^{4}}}{2 (1 - x^{2})}$

Add $1$ and $0$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{4 x^{4} + 1 - 4 x^{4}}}{2 (1 - x^{2})}$

Subtract $4 x^{4}$ from $4 x^{4}$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{0 + 1}}{2 (1 - x^{2})}$

Add $0$ and $1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1}}{2 (1 - x^{2})}$

Any root of $1$ is $1$ .

$y = \frac{1 - 2 x^{2} \pm 1}{2 (1 - x^{2})}$

Simplify the denominator.

Rewrite $1$ as $1^{2}$ .

$y = \frac{1 - 2 x^{2} \pm 1}{2 (1^{2} - x^{2})}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$y = \frac{1 - 2 x^{2} \pm 1}{2 (1 + x) (1 - x)}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Apply the distributive property.

$y = \frac{1 - (2 x^{2}) \pm \sqrt{{(- 1 + 2 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $- 1$ by $- 1$ .

$y = \frac{1 - (2 x^{2}) \pm \sqrt{{(- 1 + 2 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $2$ by $- 1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{{(- 1 + 2 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Rewrite ${(- 1 + 2 x^{2})}^{2}$ as $(- 1 + 2 x^{2}) (- 1 + 2 x^{2})$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{(- 1 + 2 x^{2}) (- 1 + 2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Expand $(- 1 + 2 x^{2}) (- 1 + 2 x^{2})$ using the FOIL Method.

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{- 1 (- 1 + 2 x^{2}) + 2 x^{2} (- 1 + 2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{- 1 \cdot - 1 - 1 (2 x^{2}) + 2 x^{2} (- 1 + 2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{- 1 \cdot - 1 - 1 (2 x^{2}) + 2 x^{2} \cdot - 1 + 2 x^{2} (2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 1$ by $- 1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 1 (2 x^{2}) + 2 x^{2} \cdot - 1 + 2 x^{2} (2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $2$ by $- 1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} + 2 x^{2} \cdot - 1 + 2 x^{2} (2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $- 1$ by $2$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 2 x^{2} (2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Rewrite using the commutative property of multiplication.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 2 \cdot (2 x^{2} x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 2 \cdot (2 (x^{2} x^{2})) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 2 \cdot (2 x^{2 + 2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Add $2$ and $2$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 2 \cdot (2 x^{4}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $2$ by $2$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 4 x^{4} - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Subtract $2 x^{2}$ from $- 2 x^{2}$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (- 4 \cdot 1 - 4 (- x^{2})) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $- 4$ by $1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (- 4 - 4 (- x^{2})) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $- 1$ by $- 4$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (- 4 + 4 x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (- 4 \cdot - 1 + 4 x^{2} \cdot - 1) x^{2}}}{2 (1 - x^{2})}$

Multiply $- 4$ by $- 1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (4 + 4 x^{2} \cdot - 1) x^{2}}}{2 (1 - x^{2})}$

Multiply $- 1$ by $4$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (4 - 4 x^{2}) x^{2}}}{2 (1 - x^{2})}$

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + 4 x^{2} - 4 x^{2} x^{2}}}{2 (1 - x^{2})}$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + 4 x^{2} - 4 (x^{2} x^{2})}}{2 (1 - x^{2})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + 4 x^{2} - 4 x^{2 + 2}}}{2 (1 - x^{2})}$

Add $2$ and $2$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + 4 x^{2} - 4 x^{4}}}{2 (1 - x^{2})}$

Add $- 4 x^{2}$ and $4 x^{2}$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 + 4 x^{4} + 0 - 4 x^{4}}}{2 (1 - x^{2})}$

Add $1$ and $0$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{4 x^{4} + 1 - 4 x^{4}}}{2 (1 - x^{2})}$

Subtract $4 x^{4}$ from $4 x^{4}$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{0 + 1}}{2 (1 - x^{2})}$

Add $0$ and $1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1}}{2 (1 - x^{2})}$

Any root of $1$ is $1$ .

$y = \frac{1 - 2 x^{2} \pm 1}{2 (1 - x^{2})}$

Simplify the denominator.

Rewrite $1$ as $1^{2}$ .

$y = \frac{1 - 2 x^{2} \pm 1}{2 (1^{2} - x^{2})}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$y = \frac{1 - 2 x^{2} \pm 1}{2 (1 + x) (1 - x)}$

Change the $\pm$ to $+$ .

$y = \frac{1 - 2 x^{2} + 1}{2 (1 + x) (1 - x)}$

Simplify the numerator.

Add $1$ and $1$ .

$y = \frac{- 2 x^{2} + 2}{2 (1 + x) (1 - x)}$

Factor $2$ out of $- 2 x^{2} + 2$ .

Factor $2$ out of $- 2 x^{2}$ .

$y = \frac{2 (- x^{2}) + 2}{2 (1 + x) (1 - x)}$

Factor $2$ out of $2$ .

$y = \frac{2 (- x^{2}) + 2 (1)}{2 (1 + x) (1 - x)}$

Factor $2$ out of $2 (- x^{2}) + 2 (1)$ .

$y = \frac{2 (- x^{2} + 1)}{2 (1 + x) (1 - x)}$

Rewrite $1$ as $1^{2}$ .

$y = \frac{2 (- x^{2} + 1^{2})}{2 (1 + x) (1 - x)}$

Reorder $- x^{2}$ and $1^{2}$ .

$y = \frac{2 (1^{2} - x^{2})}{2 (1 + x) (1 - x)}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$y = \frac{2 (1 + x) (1 - x)}{2 (1 + x) (1 - x)}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = \frac{2 (1 + x) (1 - x)}{2 (1 + x) (1 - x)}$

Rewrite the expression.

$y = \frac{(1 + x) (1 - x)}{(1 + x) (1 - x)}$

Cancel the common factor of $1 + x$ .

Cancel the common factor.

$y = \frac{(1 + x) (1 - x)}{(1 + x) (1 - x)}$

Rewrite the expression.

$y = \frac{1 - x}{1 - x}$

Cancel the common factor of $1 - x$ .

Cancel the common factor.

$y = \frac{1 - x}{1 - x}$

Rewrite the expression.

$y = 1$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Apply the distributive property.

$y = \frac{1 - (2 x^{2}) \pm \sqrt{{(- 1 + 2 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $- 1$ by $- 1$ .

$y = \frac{1 - (2 x^{2}) \pm \sqrt{{(- 1 + 2 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $2$ by $- 1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{{(- 1 + 2 x^{2})}^{2} - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Rewrite ${(- 1 + 2 x^{2})}^{2}$ as $(- 1 + 2 x^{2}) (- 1 + 2 x^{2})$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{(- 1 + 2 x^{2}) (- 1 + 2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Expand $(- 1 + 2 x^{2}) (- 1 + 2 x^{2})$ using the FOIL Method.

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{- 1 (- 1 + 2 x^{2}) + 2 x^{2} (- 1 + 2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{- 1 \cdot - 1 - 1 (2 x^{2}) + 2 x^{2} (- 1 + 2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{- 1 \cdot - 1 - 1 (2 x^{2}) + 2 x^{2} \cdot - 1 + 2 x^{2} (2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 1$ by $- 1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 1 (2 x^{2}) + 2 x^{2} \cdot - 1 + 2 x^{2} (2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $2$ by $- 1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} + 2 x^{2} \cdot - 1 + 2 x^{2} (2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $- 1$ by $2$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 2 x^{2} (2 x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Rewrite using the commutative property of multiplication.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 2 \cdot (2 x^{2} x^{2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 2 \cdot (2 (x^{2} x^{2})) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 2 \cdot (2 x^{2 + 2}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Add $2$ and $2$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 2 \cdot (2 x^{4}) - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $2$ by $2$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 2 x^{2} - 2 x^{2} + 4 x^{4} - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Subtract $2 x^{2}$ from $- 2 x^{2}$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} - 4 \cdot (1 - x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (- 4 \cdot 1 - 4 (- x^{2})) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $- 4$ by $1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (- 4 - 4 (- x^{2})) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Multiply $- 1$ by $- 4$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (- 4 + 4 x^{2}) \cdot (- 1 x^{2})}}{2 (1 - x^{2})}$

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (- 4 \cdot - 1 + 4 x^{2} \cdot - 1) x^{2}}}{2 (1 - x^{2})}$

Multiply $- 4$ by $- 1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (4 + 4 x^{2} \cdot - 1) x^{2}}}{2 (1 - x^{2})}$

Multiply $- 1$ by $4$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + (4 - 4 x^{2}) x^{2}}}{2 (1 - x^{2})}$

Apply the distributive property.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + 4 x^{2} - 4 x^{2} x^{2}}}{2 (1 - x^{2})}$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + 4 x^{2} - 4 (x^{2} x^{2})}}{2 (1 - x^{2})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + 4 x^{2} - 4 x^{2 + 2}}}{2 (1 - x^{2})}$

Add $2$ and $2$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 - 4 x^{2} + 4 x^{4} + 4 x^{2} - 4 x^{4}}}{2 (1 - x^{2})}$

Add $- 4 x^{2}$ and $4 x^{2}$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1 + 4 x^{4} + 0 - 4 x^{4}}}{2 (1 - x^{2})}$

Add $1$ and $0$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{4 x^{4} + 1 - 4 x^{4}}}{2 (1 - x^{2})}$

Subtract $4 x^{4}$ from $4 x^{4}$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{0 + 1}}{2 (1 - x^{2})}$

Add $0$ and $1$ .

$y = \frac{1 - 2 x^{2} \pm \sqrt{1}}{2 (1 - x^{2})}$

Any root of $1$ is $1$ .

$y = \frac{1 - 2 x^{2} \pm 1}{2 (1 - x^{2})}$

Simplify the denominator.

Rewrite $1$ as $1^{2}$ .

$y = \frac{1 - 2 x^{2} \pm 1}{2 (1^{2} - x^{2})}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$y = \frac{1 - 2 x^{2} \pm 1}{2 (1 + x) (1 - x)}$

Change the $\pm$ to $-$ .

$y = \frac{1 - 2 x^{2} - 1}{2 (1 + x) (1 - x)}$

Simplify the numerator.

Subtract $1$ from $1$ .

$y = \frac{- 2 x^{2} + 0}{2 (1 + x) (1 - x)}$

Add $- 2 x^{2}$ and $0$ .

$y = \frac{- 2 x^{2}}{2 (1 + x) (1 - x)}$

Cancel the common factor of $- 2$ and $2$ .

Factor $2$ out of $- 2 x^{2}$ .

$y = \frac{2 (- x^{2})}{2 (1 + x) (1 - x)}$

Cancel the common factors.

Factor $2$ out of $2 (1 + x) (1 - x)$ .

$y = \frac{2 (- x^{2})}{2 ((1 + x) (1 - x))}$

Cancel the common factor.

$y = \frac{2 (- x^{2})}{2 ((1 + x) (1 - x))}$

Rewrite the expression.

$y = \frac{- x^{2}}{(1 + x) (1 - x)}$

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{(1 + x) (1 - x)}$

The final answer is the combination of both solutions.

$y = 1$

$y = - \frac{x^{2}}{(1 + x) (1 - x)}$

$y = 1$

$y = - \frac{x^{2}}{(1 + x) (1 - x)}$

$y = 1$

$y = - \frac{x^{2}}{(1 + x) (1 - x)}$

$y = 1$

$y = - \frac{x^{2}}{(1 + x) (1 - x)}$

Remove the numeric answers.

$y = - \frac{x^{2}}{(1 + x) (1 - x)}$

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = - \frac{x^{2}}{(1 + x) (1 - x)}$

Verify if $f^{- 1} (x) = - \frac{x^{2}}{(1 + x) (1 - x)}$ is the inverse of $f (x) = \sqrt{\frac{x}{x - 1}}$ .

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Evaluate $f^{- 1} (f (x))$ .

Set up the composite result function.

$f^{- 1} (f (x))$

Evaluate $f^{- 1} (\sqrt{\frac{x}{x - 1}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{{(\sqrt{\frac{x}{x - 1}})}^{2}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - (\sqrt{\frac{x}{x - 1}}))}$

Remove parentheses.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{{(\sqrt{\frac{x}{x - 1}})}^{2}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - (\sqrt{\frac{x}{x - 1}}))}$

Simplify the numerator.

Rewrite $\sqrt{\frac{x}{x - 1}}$ as $\frac{\sqrt{x}}{\sqrt{x - 1}}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{{(\frac{\sqrt{x}}{\sqrt{x - 1}})}^{2}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Multiply $\frac{\sqrt{x}}{\sqrt{x - 1}}$ by $\frac{\sqrt{x - 1}}{\sqrt{x - 1}}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{{(\frac{\sqrt{x}}{\sqrt{x - 1}} \cdot \frac{\sqrt{x - 1}}{\sqrt{x - 1}})}^{2}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{x}}{\sqrt{x - 1}}$ by $\frac{\sqrt{x - 1}}{\sqrt{x - 1}}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{{(\frac{\sqrt{x} \sqrt{x - 1}}{\sqrt{x - 1} \sqrt{x - 1}})}^{2}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Raise $\sqrt{x - 1}$ to the power of $1$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{{(\frac{\sqrt{x} \sqrt{x - 1}}{\sqrt{x - 1} \sqrt{x - 1}})}^{2}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Raise $\sqrt{x - 1}$ to the power of $1$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{{(\frac{\sqrt{x} \sqrt{x - 1}}{\sqrt{x - 1} \sqrt{x - 1}})}^{2}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{{(\frac{\sqrt{x} \sqrt{x - 1}}{{\sqrt{x - 1}}^{1 + 1}})}^{2}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Add $1$ and $1$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{{(\frac{\sqrt{x} \sqrt{x - 1}}{{\sqrt{x - 1}}^{2}})}^{2}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Rewrite ${\sqrt{x - 1}}^{2}$ as $x - 1$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 1}$ as ${(x - 1)}^{\frac{1}{2}}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{{(\frac{\sqrt{x} \sqrt{x - 1}}{{({(x - 1)}^{\frac{1}{2}})}^{2}})}^{2}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{{(\frac{\sqrt{x} \sqrt{x - 1}}{{(x - 1)}^{\frac{1}{2} \cdot 2}})}^{2}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Combine $\frac{1}{2}$ and $2$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{{(\frac{\sqrt{x} \sqrt{x - 1}}{{(x - 1)}^{\frac{2}{2}}})}^{2}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{{(\frac{\sqrt{x} \sqrt{x - 1}}{{(x - 1)}^{\frac{2}{2}}})}^{2}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Rewrite the expression.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{{(\frac{\sqrt{x} \sqrt{x - 1}}{x - 1})}^{2}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Simplify.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{{(\frac{\sqrt{x} \sqrt{x - 1}}{x - 1})}^{2}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Combine using the product rule for radicals.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{{(\frac{\sqrt{x (x - 1)}}{x - 1})}^{2}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Apply the product rule to $\frac{\sqrt{x (x - 1)}}{x - 1}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{{\sqrt{x (x - 1)}}^{2}}{{(x - 1)}^{2}}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Simplify the numerator.

Rewrite ${\sqrt{x (x - 1)}}^{2}$ as $x (x - 1)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x (x - 1)}$ as ${(x (x - 1))}^{\frac{1}{2}}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{{({(x (x - 1))}^{\frac{1}{2}})}^{2}}{{(x - 1)}^{2}}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{{(x (x - 1))}^{\frac{1}{2} \cdot 2}}{{(x - 1)}^{2}}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Combine $\frac{1}{2}$ and $2$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{{(x (x - 1))}^{\frac{2}{2}}}{{(x - 1)}^{2}}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{{(x (x - 1))}^{\frac{2}{2}}}{{(x - 1)}^{2}}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Rewrite the expression.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x (x - 1)}{{(x - 1)}^{2}}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Simplify.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x (x - 1)}{{(x - 1)}^{2}}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Apply the distributive property.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x \cdot x + x \cdot - 1}{{(x - 1)}^{2}}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Multiply $x$ by $x$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x^{2} + x \cdot - 1}{{(x - 1)}^{2}}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Move $- 1$ to the left of $x$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x^{2} - 1 \cdot x}{{(x - 1)}^{2}}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Rewrite $- 1 x$ as $- x$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x^{2} - x}{{(x - 1)}^{2}}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Factor $x$ out of $x^{2} - x$ .

Factor $x$ out of $x^{2}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x \cdot x - x}{{(x - 1)}^{2}}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Factor $x$ out of $- x$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x \cdot x + x \cdot - 1}{{(x - 1)}^{2}}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x (x - 1)}{{(x - 1)}^{2}}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Cancel the common factor of $x - 1$ and ${(x - 1)}^{2}$ .

Factor $x - 1$ out of $x (x - 1)$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{(x - 1) x}{{(x - 1)}^{2}}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Cancel the common factors.

Factor $x - 1$ out of ${(x - 1)}^{2}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{(x - 1) x}{(x - 1) (x - 1)}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Cancel the common factor.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{(x - 1) x}{(x - 1) (x - 1)}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Rewrite the expression.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{(1 + \sqrt{\frac{x}{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Simplify the denominator.

Rewrite $\sqrt{\frac{x}{x - 1}}$ as $\frac{\sqrt{x}}{\sqrt{x - 1}}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{(1 + \frac{\sqrt{x}}{\sqrt{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Multiply $\frac{\sqrt{x}}{\sqrt{x - 1}}$ by $\frac{\sqrt{x - 1}}{\sqrt{x - 1}}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{(1 + \frac{\sqrt{x}}{\sqrt{x - 1}} \cdot \frac{\sqrt{x - 1}}{\sqrt{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{x}}{\sqrt{x - 1}}$ by $\frac{\sqrt{x - 1}}{\sqrt{x - 1}}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{(1 + \frac{\sqrt{x} \sqrt{x - 1}}{\sqrt{x - 1} \sqrt{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Raise $\sqrt{x - 1}$ to the power of $1$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{(1 + \frac{\sqrt{x} \sqrt{x - 1}}{\sqrt{x - 1} \sqrt{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Raise $\sqrt{x - 1}$ to the power of $1$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{(1 + \frac{\sqrt{x} \sqrt{x - 1}}{\sqrt{x - 1} \sqrt{x - 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{(1 + \frac{\sqrt{x} \sqrt{x - 1}}{{\sqrt{x - 1}}^{1 + 1}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Add $1$ and $1$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{(1 + \frac{\sqrt{x} \sqrt{x - 1}}{{\sqrt{x - 1}}^{2}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Rewrite ${\sqrt{x - 1}}^{2}$ as $x - 1$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 1}$ as ${(x - 1)}^{\frac{1}{2}}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{(1 + \frac{\sqrt{x} \sqrt{x - 1}}{{({(x - 1)}^{\frac{1}{2}})}^{2}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{(1 + \frac{\sqrt{x} \sqrt{x - 1}}{{(x - 1)}^{\frac{1}{2} \cdot 2}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Combine $\frac{1}{2}$ and $2$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{(1 + \frac{\sqrt{x} \sqrt{x - 1}}{{(x - 1)}^{\frac{2}{2}}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{(1 + \frac{\sqrt{x} \sqrt{x - 1}}{{(x - 1)}^{\frac{2}{2}}}) (1 - \sqrt{\frac{x}{x - 1}})}$

Rewrite the expression.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{(1 + \frac{\sqrt{x} \sqrt{x - 1}}{x - 1}) (1 - \sqrt{\frac{x}{x - 1}})}$

Simplify.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{(1 + \frac{\sqrt{x} \sqrt{x - 1}}{x - 1}) (1 - \sqrt{\frac{x}{x - 1}})}$

Combine using the product rule for radicals.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{(1 + \frac{\sqrt{x (x - 1)}}{x - 1}) (1 - \sqrt{\frac{x}{x - 1}})}$

Write $1$ as a fraction with a common denominator.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{(\frac{x - 1}{x - 1} + \frac{\sqrt{x (x - 1)}}{x - 1}) (1 - \sqrt{\frac{x}{x - 1}})}$

Combine the numerators over the common denominator.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot (1 - \sqrt{\frac{x}{x - 1}})}$

Rewrite $\sqrt{\frac{x}{x - 1}}$ as $\frac{\sqrt{x}}{\sqrt{x - 1}}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot (1 - \frac{\sqrt{x}}{\sqrt{x - 1}})}$

Multiply $\frac{\sqrt{x}}{\sqrt{x - 1}}$ by $\frac{\sqrt{x - 1}}{\sqrt{x - 1}}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot (1 - (\frac{\sqrt{x}}{\sqrt{x - 1}} \cdot \frac{\sqrt{x - 1}}{\sqrt{x - 1}}))}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{x}}{\sqrt{x - 1}}$ by $\frac{\sqrt{x - 1}}{\sqrt{x - 1}}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot (1 - \frac{\sqrt{x} \sqrt{x - 1}}{\sqrt{x - 1} \sqrt{x - 1}})}$

Raise $\sqrt{x - 1}$ to the power of $1$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot (1 - \frac{\sqrt{x} \sqrt{x - 1}}{\sqrt{x - 1} \sqrt{x - 1}})}$

Raise $\sqrt{x - 1}$ to the power of $1$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot (1 - \frac{\sqrt{x} \sqrt{x - 1}}{\sqrt{x - 1} \sqrt{x - 1}})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot (1 - \frac{\sqrt{x} \sqrt{x - 1}}{{\sqrt{x - 1}}^{1 + 1}})}$

Add $1$ and $1$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot (1 - \frac{\sqrt{x} \sqrt{x - 1}}{{\sqrt{x - 1}}^{2}})}$

Rewrite ${\sqrt{x - 1}}^{2}$ as $x - 1$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 1}$ as ${(x - 1)}^{\frac{1}{2}}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot (1 - \frac{\sqrt{x} \sqrt{x - 1}}{{({(x - 1)}^{\frac{1}{2}})}^{2}})}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot (1 - \frac{\sqrt{x} \sqrt{x - 1}}{{(x - 1)}^{\frac{1}{2} \cdot 2}})}$

Combine $\frac{1}{2}$ and $2$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot (1 - \frac{\sqrt{x} \sqrt{x - 1}}{{(x - 1)}^{\frac{2}{2}}})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot (1 - \frac{\sqrt{x} \sqrt{x - 1}}{{(x - 1)}^{\frac{2}{2}}})}$

Rewrite the expression.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot (1 - \frac{\sqrt{x} \sqrt{x - 1}}{x - 1})}$

Simplify.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot (1 - \frac{\sqrt{x} \sqrt{x - 1}}{x - 1})}$

Combine using the product rule for radicals.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot (1 - \frac{\sqrt{x (x - 1)}}{x - 1})}$

Write $1$ as a fraction with a common denominator.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot (\frac{x - 1}{x - 1} - \frac{\sqrt{x (x - 1)}}{x - 1})}$

Combine the numerators over the common denominator.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1} \cdot \frac{x - 1 - \sqrt{x (x - 1)}}{x - 1}}$

Multiply $\frac{x - 1 + \sqrt{x (x - 1)}}{x - 1}$ by $\frac{x - 1 - \sqrt{x (x - 1)}}{x - 1}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{(x - 1 + \sqrt{x (x - 1)}) (x - 1 - \sqrt{x (x - 1)})}{(x - 1) (x - 1)}}$

Simplify the denominator.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{(x - 1 + \sqrt{x (x - 1)}) (x - 1 - \sqrt{x (x - 1)})}{{(x - 1)}^{1 + 1}}}$

Add $1$ and $1$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{\frac{x}{x - 1}}{\frac{(x - 1 + \sqrt{x (x - 1)}) (x - 1 - \sqrt{x (x - 1)})}{{(x - 1)}^{2}}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - (\frac{x}{x - 1} \cdot \frac{{(x - 1)}^{2}}{(x - 1 + \sqrt{x (x - 1)}) (x - 1 - \sqrt{x (x - 1)})})$

Cancel the common factor of $x - 1$ .

Factor $x - 1$ out of ${(x - 1)}^{2}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - (\frac{x}{x - 1} \cdot \frac{(x - 1) (x - 1)}{(x - 1 + \sqrt{x (x - 1)}) (x - 1 - \sqrt{x (x - 1)})})$

Cancel the common factor.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - (\frac{x}{x - 1} \cdot \frac{(x - 1) (x - 1)}{(x - 1 + \sqrt{x (x - 1)}) (x - 1 - \sqrt{x (x - 1)})})$

Rewrite the expression.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - (x (\frac{x - 1}{(x - 1 + \sqrt{x (x - 1)}) (x - 1 - \sqrt{x (x - 1)})}))$

Combine $x$ and $\frac{x - 1}{(x - 1 + \sqrt{x (x - 1)}) (x - 1 - \sqrt{x (x - 1)})}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{x (x - 1)}{(x - 1 + \sqrt{x (x - 1)}) (x - 1 - \sqrt{x (x - 1)})}$

Multiply $\frac{x (x - 1)}{(x - 1 + \sqrt{x (x - 1)}) (x - 1 - \sqrt{x (x - 1)})}$ by $\frac{x - 1 - \sqrt{x (x - 1)}}{x - 1 - \sqrt{x (x - 1)}}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - (\frac{x (x - 1)}{(x - 1 + \sqrt{x (x - 1)}) (x - 1 - \sqrt{x (x - 1)})} \cdot \frac{x - 1 - \sqrt{x (x - 1)}}{x - 1 - \sqrt{x (x - 1)}})$

Multiply $\frac{x (x - 1)}{(x - 1 + \sqrt{x (x - 1)}) (x - 1 - \sqrt{x (x - 1)})}$ by $\frac{x - 1 - \sqrt{x (x - 1)}}{x - 1 - \sqrt{x (x - 1)}}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{x (x - 1) (x - 1 - \sqrt{x (x - 1)})}{(x - 1 + \sqrt{x (x - 1)}) (x - 1 - \sqrt{x (x - 1)}) (x - 1 - \sqrt{x (x - 1)})}$

Reorder.

Move $x - 1 + \sqrt{x (x - 1)}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{x (x - 1) (x - 1 - \sqrt{x (x - 1)})}{(x - 1 - \sqrt{x (x - 1)}) ((x - 1 + \sqrt{x (x - 1)}) (x - 1 - \sqrt{x (x - 1)}))}$

Expand the denominator using the FOIL method.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{x (x - 1) (x - 1 - \sqrt{x (x - 1)})}{(x - 1 - \sqrt{x (x - 1)}) (x^{2} + x \cdot - 1 + x (- \sqrt{x (x - 1)}) - x + 1 + \sqrt{x (x - 1)} + \sqrt{x (x - 1)} x + \sqrt{x (x - 1)} \cdot - 1 - {\sqrt{x (x - 1)}}^{2})}$

Simplify.

Subtract $x^{2}$ from $x^{2}$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{x (x - 1) (x - 1 - \sqrt{x (x - 1)})}{(x - 1 - \sqrt{x (x - 1)}) (- x - x + 1 + 0 + x)}$

Add $- x - x + 1$ and $0$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{x (x - 1) (x - 1 - \sqrt{x (x - 1)})}{(x - 1 - \sqrt{x (x - 1)}) (- x - x + 1 + x)}$

Add $- x$ and $x$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{x (x - 1) (x - 1 - \sqrt{x (x - 1)})}{(x - 1 - \sqrt{x (x - 1)}) (- x + 1 + 0)}$

Add $- x + 1$ and $0$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{x (x - 1) (x - 1 - \sqrt{x (x - 1)})}{(x - 1 - \sqrt{x (x - 1)}) (- x + 1)}$

Cancel the common factor of $x - 1 - \sqrt{x (x - 1)}$ .

Cancel the common factor.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{x (x - 1) (x - 1 - \sqrt{x (x - 1)})}{(x - 1 - \sqrt{x (x - 1)}) (- x + 1)}$

Rewrite the expression.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{x (x - 1)}{- x + 1}$

Cancel the common factor of $x - 1$ and $- x + 1$ .

Factor $- 1$ out of $x$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{x (- 1 (- x) - 1)}{- x + 1}$

Rewrite $- 1$ as $- 1 (1)$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{x (- 1 (- x) - 1 \cdot 1)}{- x + 1}$

Factor $- 1$ out of $- 1 (- x) - 1 (1)$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{x (- 1 (- x + 1))}{- x + 1}$

Cancel the common factor.

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - \frac{x (- 1 (- x + 1))}{- x + 1}$

Divide $x \cdot (- 1)$ by $1$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - (x \cdot (- 1))$

Simplify the expression.

Move $- 1$ to the left of $x$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = - (- 1 x)$

Rewrite $- 1 x$ as $- x$ .

$f^{- 1} (\sqrt{\frac{x}{x - 1}}) = x$

Evaluate $f (f^{- 1} (x))$ .

Set up the composite result function.

$f (f^{- 1} (x))$

Evaluate $f (- \frac{x^{2}}{(1 + x) (1 - x)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{- \frac{x^{2}}{(1 + x) (1 - x)}}{(- \frac{x^{2}}{(1 + x) (1 - x)}) - 1}}$

Factor $- 1$ out of $- \frac{x^{2}}{(1 + x) (1 - x)} - 1$ .

Reorder the expression.

Reorder $1$ and $x$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{- \frac{x^{2}}{(1 + x) (1 - x)}}{- \frac{x^{2}}{(x + 1) (1 - x)} - 1}}$

Reorder $1$ and $- x$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{- \frac{x^{2}}{(1 + x) (1 - x)}}{- \frac{x^{2}}{(x + 1) (- x + 1)} - 1}}$

Factor $- 1$ out of $- \frac{x^{2}}{(x + 1) (- x + 1)}$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{- \frac{x^{2}}{(1 + x) (1 - x)}}{- (\frac{x^{2}}{(x + 1) (- x + 1)}) - 1}}$

Rewrite $- 1$ as $- 1 (1)$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{- \frac{x^{2}}{(1 + x) (1 - x)}}{- (\frac{x^{2}}{(x + 1) (- x + 1)}) - 1 \cdot 1}}$

Factor $- 1$ out of $- (\frac{x^{2}}{(x + 1) (- x + 1)}) - 1 (1)$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{- \frac{x^{2}}{(1 + x) (1 - x)}}{- (\frac{x^{2}}{(x + 1) (- x + 1)} + 1)}}$

Rewrite the expression.

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{\frac{x^{2}}{(1 + x) (1 - x)}}{\frac{x^{2}}{(x + 1) (- x + 1)} + 1}}$

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x)} \cdot \frac{1}{\frac{x^{2}}{(x + 1) (- x + 1)} + 1}}$

Simplify the denominator.

Write $1$ as a fraction with a common denominator.

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x)} \cdot \frac{1}{\frac{x^{2}}{(x + 1) (- x + 1)} + \frac{(x + 1) (- x + 1)}{(x + 1) (- x + 1)}}}$

Combine the numerators over the common denominator.

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x)} \cdot \frac{1}{\frac{x^{2} + (x + 1) (- x + 1)}{(x + 1) (- x + 1)}}}$

Reorder terms.

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x)} \cdot \frac{1}{\frac{x^{2} + (- x + 1) (x + 1)}{(- x + 1) (x + 1)}}}$

Rewrite $\frac{x^{2} + (- x + 1) (x + 1)}{(- x + 1) (x + 1)}$ in a factored form.

Expand $(- x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x)} \cdot \frac{1}{\frac{x^{2} - x (x + 1) + 1 (x + 1)}{(- x + 1) (x + 1)}}}$

Apply the distributive property.

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x)} \cdot \frac{1}{\frac{x^{2} - x \cdot x - x \cdot 1 + 1 (x + 1)}{(- x + 1) (x + 1)}}}$

Apply the distributive property.

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x)} \cdot \frac{1}{\frac{x^{2} - x \cdot x - x \cdot 1 + 1 x + 1 \cdot 1}{(- x + 1) (x + 1)}}}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x)} \cdot \frac{1}{\frac{x^{2} - (x \cdot x) - x \cdot 1 + 1 x + 1 \cdot 1}{(- x + 1) (x + 1)}}}$

Multiply $x$ by $x$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x)} \cdot \frac{1}{\frac{x^{2} - x^{2} - x \cdot 1 + 1 x + 1 \cdot 1}{(- x + 1) (x + 1)}}}$

Multiply $- 1$ by $1$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x)} \cdot \frac{1}{\frac{x^{2} - x^{2} - x + 1 x + 1 \cdot 1}{(- x + 1) (x + 1)}}}$

Multiply $x$ by $1$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x)} \cdot \frac{1}{\frac{x^{2} - x^{2} - x + x + 1 \cdot 1}{(- x + 1) (x + 1)}}}$

Multiply $1$ by $1$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x)} \cdot \frac{1}{\frac{x^{2} - x^{2} - x + x + 1}{(- x + 1) (x + 1)}}}$

Add $- x$ and $x$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x)} \cdot \frac{1}{\frac{x^{2} - x^{2} + 0 + 1}{(- x + 1) (x + 1)}}}$

Add $- x^{2}$ and $0$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x)} \cdot \frac{1}{\frac{x^{2} - x^{2} + 1}{(- x + 1) (x + 1)}}}$

Subtract $x^{2}$ from $x^{2}$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x)} \cdot \frac{1}{\frac{0 + 1}{(- x + 1) (x + 1)}}}$

Add $0$ and $1$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x)} \cdot \frac{1}{\frac{1}{(- x + 1) (x + 1)}}}$

Combine fractions.

Combine.

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2} \cdot 1}{(1 + x) (1 - x) (\frac{1}{(- x + 1) (x + 1)})}}$

Multiply $x^{2}$ by $1$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{x^{2}}{(1 + x) (1 - x) (\frac{1}{(- x + 1) (x + 1)})}}$

Factor $\frac{1}{(- x + 1) (x + 1)}$ out of $\frac{x^{2}}{(1 + x) (1 - x) \frac{1}{(- x + 1) (x + 1)}}$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{(- x + 1) (x + 1) (\frac{x^{2}}{(1 + x) (1 - x)})}$

Expand $(- x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{(- x (x + 1) + 1 (x + 1)) (\frac{x^{2}}{(1 + x) (1 - x)})}$

Apply the distributive property.

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{(- x \cdot x - x \cdot 1 + 1 (x + 1)) (\frac{x^{2}}{(1 + x) (1 - x)})}$

Apply the distributive property.

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{(- x \cdot x - x \cdot 1 + 1 x + 1 \cdot 1) (\frac{x^{2}}{(1 + x) (1 - x)})}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{(- (x \cdot x) - x \cdot 1 + 1 x + 1 \cdot 1) (\frac{x^{2}}{(1 + x) (1 - x)})}$

Multiply $x$ by $x$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{(- x^{2} - x \cdot 1 + 1 x + 1 \cdot 1) (\frac{x^{2}}{(1 + x) (1 - x)})}$

Multiply $- 1$ by $1$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{(- x^{2} - x + 1 x + 1 \cdot 1) (\frac{x^{2}}{(1 + x) (1 - x)})}$

Multiply $x$ by $1$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{(- x^{2} - x + x + 1 \cdot 1) (\frac{x^{2}}{(1 + x) (1 - x)})}$

Multiply $1$ by $1$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{(- x^{2} - x + x + 1) (\frac{x^{2}}{(1 + x) (1 - x)})}$

Add $- x$ and $x$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{(- x^{2} + 0 + 1) (\frac{x^{2}}{(1 + x) (1 - x)})}$

Add $- x^{2}$ and $0$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{(- x^{2} + 1) (\frac{x^{2}}{(1 + x) (1 - x)})}$

Multiply $- x^{2} + 1$ by $\frac{x^{2}}{(1 + x) (1 - x)}$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{(- x^{2} + 1) x^{2}}{(1 + x) (1 - x)}}$

Simplify the numerator.

Rewrite $1$ as $1^{2}$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{(- x^{2} + 1^{2}) x^{2}}{(1 + x) (1 - x)}}$

Reorder $- x^{2}$ and $1^{2}$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{(1^{2} - x^{2}) x^{2}}{(1 + x) (1 - x)}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{(1 + x) (1 - x) x^{2}}{(1 + x) (1 - x)}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $1 + x$ .

Cancel the common factor.

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{(1 + x) (1 - x) x^{2}}{(1 + x) (1 - x)}}$

Rewrite the expression.

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{(1 - x) x^{2}}{1 - x}}$

Cancel the common factor of $1 - x$ .

Cancel the common factor.

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{\frac{(1 - x) x^{2}}{1 - x}}$

Divide $x^{2}$ by $1$ .

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = \sqrt{x^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$f (- \frac{x^{2}}{(1 + x) (1 - x)}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{x^{2}}{(1 + x) (1 - x)}$ is the inverse of $f (x) = \sqrt{\frac{x}{x - 1}}$ .

$f^{- 1} (x) = - \frac{x^{2}}{(1 + x) (1 - x)}$

Find the Inverse f(x) = square root of x/(x-1)

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