Find the Inverse f(x)=x(x-1)

$f (x) = x (x - 1)$

Step 1

Write $f (x) = x (x - 1)$ as an equation.

$y = x (x - 1)$

Step 2

Interchange the variables.

$x = y (y - 1)$

Step 3

Solve for $y$ .

Rewrite the equation as $y (y - 1) = x$ .

$y (y - 1) = x$

Simplify $y (y - 1)$ .

Simplify by multiplying through.

Apply the distributive property.

$y \cdot y + y \cdot - 1 = x$

Simplify the expression.

Multiply $y$ by $y$ .

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

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

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

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

$y^{2} - y = x$

Subtract $x$ from both sides of the equation.

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

Factor $- 1$ out of ${(- 1)}^{2} - 4 \cdot 1 \cdot - 1 x$ .

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

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

Factor $- 1$ out of $- 4 \cdot 1 \cdot - 1 x$ .

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

Factor $- 1$ out of $- - 1 - (- 4 \cdot 1 x)$ .

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

Multiply $- 4$ by $1$ .

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

Multiply $2$ by $1$ .

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

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

Simplify the numerator.

Factor $- 1$ out of ${(- 1)}^{2} - 4 \cdot 1 \cdot - 1 x$ .

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

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

Factor $- 1$ out of $- 4 \cdot 1 \cdot - 1 x$ .

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

Factor $- 1$ out of $- - 1 - (- 4 \cdot 1 x)$ .

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

Multiply $- 4$ by $1$ .

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

Multiply $2$ by $1$ .

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

Change the $\pm$ to $+$ .

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

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

Simplify the numerator.

Factor $- 1$ out of ${(- 1)}^{2} - 4 \cdot 1 \cdot - 1 x$ .

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

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

Factor $- 1$ out of $- 4 \cdot 1 \cdot - 1 x$ .

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

Factor $- 1$ out of $- - 1 - (- 4 \cdot 1 x)$ .

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

Multiply $- 4$ by $1$ .

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

Multiply $2$ by $1$ .

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

Change the $\pm$ to $-$ .

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

The final answer is the combination of both solutions.

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

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

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

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

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = \frac{1 + \sqrt{- (- 1 - 4 x)}}{2}, \frac{1 - \sqrt{- (- 1 - 4 x)}}{2}$ is the inverse of $f (x) = x (x - 1)$ .

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = x (x - 1)$ and $f^{- 1} (x) = \frac{1 + \sqrt{- (- 1 - 4 x)}}{2}, \frac{1 - \sqrt{- (- 1 - 4 x)}}{2}$ and compare them.

Find the range of $f (x) = x (x - 1)$ .

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[- \frac{1}{4}, \infty)$

Find the domain of $\frac{1 + \sqrt{- (- 1 - 4 x)}}{2}$ .

Set the radicand in $\sqrt{- (- 1 - 4 x)}$ greater than or equal to $0$ to find where the expression is defined.

$- (- 1 - 4 x) \geq 0$

Solve for $x$ .

Simplify $- (- 1 - 4 x)$ .

Apply the distributive property.

$- - 1 - (- 4 x) \geq 0$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 - (- 4 x) \geq 0$

Multiply $- 4$ by $- 1$ .

$1 + 4 x \geq 0$

Subtract $1$ from both sides of the inequality.

$4 x \geq - 1$

Divide each term in $4 x \geq - 1$ by $4$ and simplify.

Divide each term in $4 x \geq - 1$ by $4$ .

$\frac{4 x}{4} \geq \frac{- 1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{- 1}{4}$

Divide $x$ by $1$ .

$x \geq \frac{- 1}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x \geq - \frac{1}{4}$

The domain is all values of $x$ that make the expression defined.

$[- \frac{1}{4}, \infty)$

Find the domain of $f (x) = x (x - 1)$ .

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

$(- \infty, \infty)$

Since the domain of $f^{- 1} (x) = \frac{1 + \sqrt{- (- 1 - 4 x)}}{2}, \frac{1 - \sqrt{- (- 1 - 4 x)}}{2}$ is the range of $f (x) = x (x - 1)$ and the range of $f^{- 1} (x) = \frac{1 + \sqrt{- (- 1 - 4 x)}}{2}, \frac{1 - \sqrt{- (- 1 - 4 x)}}{2}$ is the domain of $f (x) = x (x - 1)$ , then $f^{- 1} (x) = \frac{1 + \sqrt{- (- 1 - 4 x)}}{2}, \frac{1 - \sqrt{- (- 1 - 4 x)}}{2}$ is the inverse of $f (x) = x (x - 1)$ .

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

Step 6

Find the Inverse f(x)=x(x-1)

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