Find the Inverse h(x)=-2/(x-1)-3

$h (x) = - \frac{2}{x - 1} - 3$

Write $h (x) = - \frac{2}{x - 1} - 3$ as an equation.

$y = - \frac{2}{x - 1} - 3$

Interchange the variables.

$x = - \frac{2}{y - 1} - 3$

Solve for $y$ .

Rewrite the equation as $- \frac{2}{y - 1} - 3 = x$ .

$- \frac{2}{y - 1} - 3 = x$

Add $3$ to both sides of the equation.

$- \frac{2}{y - 1} = x + 3$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y - 1, 1, 1$

Remove parentheses.

$y - 1, 1, 1$

The LCM of one and any expression is the expression.

$y - 1$

Multiply each term in $- \frac{2}{y - 1} = x + 3$ by $y - 1$ to eliminate the fractions.

Multiply each term in $- \frac{2}{y - 1} = x + 3$ by $y - 1$ .

$- \frac{2}{y - 1} (y - 1) = x (y - 1) + 3 (y - 1)$

Simplify the left side.

Cancel the common factor of $y - 1$ .

Move the leading negative in $- \frac{2}{y - 1}$ into the numerator.

$\frac{- 2}{y - 1} (y - 1) = x (y - 1) + 3 (y - 1)$

Cancel the common factor.

$\frac{- 2}{y - 1} (y - 1) = x (y - 1) + 3 (y - 1)$

Rewrite the expression.

$- 2 = x (y - 1) + 3 (y - 1)$

Simplify the right side.

Simplify each term.

Apply the distributive property.

$- 2 = x y + x \cdot - 1 + 3 (y - 1)$

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

$- 2 = x y - 1 \cdot x + 3 (y - 1)$

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

$- 2 = x y - x + 3 (y - 1)$

Apply the distributive property.

$- 2 = x y - x + 3 y + 3 \cdot - 1$

Multiply $3$ by $- 1$ .

$- 2 = x y - x + 3 y - 3$

Solve the equation.

Rewrite the equation as $x y - x + 3 y - 3 = - 2$ .

$x y - x + 3 y - 3 = - 2$

Move all terms not containing $y$ to the right side of the equation.

Add $x$ to both sides of the equation.

$x y + 3 y - 3 = - 2 + x$

Add $3$ to both sides of the equation.

$x y + 3 y = - 2 + x + 3$

Add $- 2$ and $3$ .

$x y + 3 y = x + 1$

Factor $y$ out of $x y + 3 y$ .

Factor $y$ out of $x y$ .

$y x + 3 y = x + 1$

Factor $y$ out of $3 y$ .

$y x + y \cdot 3 = x + 1$

Factor $y$ out of $y x + y \cdot 3$ .

$y (x + 3) = x + 1$

Divide each term in $y (x + 3) = x + 1$ by $x + 3$ and simplify.

Divide each term in $y (x + 3) = x + 1$ by $x + 3$ .

$\frac{y (x + 3)}{x + 3} = \frac{x}{x + 3} + \frac{1}{x + 3}$

Simplify the left side.

Cancel the common factor of $x + 3$ .

Cancel the common factor.

$\frac{y (x + 3)}{x + 3} = \frac{x}{x + 3} + \frac{1}{x + 3}$

Divide $y$ by $1$ .

$y = \frac{x}{x + 3} + \frac{1}{x + 3}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{x + 1}{x + 3}$

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

$h^{- 1} (x) = \frac{x + 1}{x + 3}$

Verify if $h^{- 1} (x) = \frac{x + 1}{x + 3}$ is the inverse of $h (x) = - \frac{2}{x - 1} - 3$ .

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

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

Set up the composite result function.

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

Evaluate $h^{- 1} (- \frac{2}{x - 1} - 3)$ by substituting in the value of $h$ into $h^{- 1}$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{(- \frac{2}{x - 1} - 3) + 1}{(- \frac{2}{x - 1} - 3) + 3}$

Multiply the numerator and denominator of the complex fraction by $x - 1$ .

Multiply $\frac{- \frac{2}{x - 1} - 3 + 1}{- \frac{2}{x - 1} - 3 + 3}$ by $\frac{x - 1}{x - 1}$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{x - 1}{x - 1} \cdot \frac{- \frac{2}{x - 1} - 3 + 1}{- \frac{2}{x - 1} - 3 + 3}$

Combine.

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{(x - 1) (- \frac{2}{x - 1} - 3 + 1)}{(x - 1) (- \frac{2}{x - 1} - 3 + 3)}$

Apply the distributive property.

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{(x - 1) (- \frac{2}{x - 1}) + (x - 1) \cdot - 3 + (x - 1) \cdot 1}{(x - 1) (- \frac{2}{x - 1}) + (x - 1) \cdot - 3 + (x - 1) \cdot 3}$

Simplify by cancelling.

Cancel the common factor of $x - 1$ .

Move the leading negative in $- \frac{2}{x - 1}$ into the numerator.

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{(x - 1) (\frac{- 2}{x - 1}) + (x - 1) \cdot - 3 + (x - 1) \cdot 1}{(x - 1) (- \frac{2}{x - 1}) + (x - 1) \cdot - 3 + (x - 1) \cdot 3}$

Cancel the common factor.

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{(x - 1) (\frac{- 2}{x - 1}) + (x - 1) \cdot - 3 + (x - 1) \cdot 1}{(x - 1) (- \frac{2}{x - 1}) + (x - 1) \cdot - 3 + (x - 1) \cdot 3}$

Rewrite the expression.

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 + (x - 1) \cdot - 3 + (x - 1) \cdot 1}{(x - 1) (- \frac{2}{x - 1}) + (x - 1) \cdot - 3 + (x - 1) \cdot 3}$

Cancel the common factor of $x - 1$ .

Move the leading negative in $- \frac{2}{x - 1}$ into the numerator.

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 + (x - 1) \cdot - 3 + (x - 1) \cdot 1}{(x - 1) (\frac{- 2}{x - 1}) + (x - 1) \cdot - 3 + (x - 1) \cdot 3}$

Cancel the common factor.

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 + (x - 1) \cdot - 3 + (x - 1) \cdot 1}{(x - 1) (\frac{- 2}{x - 1}) + (x - 1) \cdot - 3 + (x - 1) \cdot 3}$

Rewrite the expression.

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 + (x - 1) \cdot - 3 + (x - 1) \cdot 1}{- 2 + (x - 1) \cdot - 3 + (x - 1) \cdot 3}$

Simplify the numerator.

Reorder $x - 1$ and $1$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 - 3 \cdot (x - 1) + 1 \cdot (x - 1)}{- 2 + (x - 1) \cdot - 3 + (x - 1) \cdot 3}$

Add $- 3 \cdot (x - 1)$ and $1 \cdot (x - 1)$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 - 2 \cdot (x - 1)}{- 2 + (x - 1) \cdot - 3 + (x - 1) \cdot 3}$

Factor $- 2$ out of $- 2 - 2 \cdot (x - 1)$ .

Reorder $- 2$ and $- 2 \cdot (x - 1)$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 (x - 1) - 2}{- 2 + (x - 1) \cdot - 3 + (x - 1) \cdot 3}$

Factor $- 2$ out of $- 2$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 (x - 1) - 2 \cdot 1}{- 2 + (x - 1) \cdot - 3 + (x - 1) \cdot 3}$

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

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 (x - 1 + 1)}{- 2 + (x - 1) \cdot - 3 + (x - 1) \cdot 3}$

Add $- 1$ and $1$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 (x + 0)}{- 2 + (x - 1) \cdot - 3 + (x - 1) \cdot 3}$

Add $x$ and $0$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 x}{- 2 + (x - 1) \cdot - 3 + (x - 1) \cdot 3}$

Simplify the denominator.

Apply the distributive property.

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 x}{- 2 + x \cdot - 3 - 1 \cdot - 3 + (x - 1) \cdot 3}$

Move $- 3$ to the left of $x$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 x}{- 2 - 3 \cdot x - 1 \cdot - 3 + (x - 1) \cdot 3}$

Multiply $- 1$ by $- 3$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 x}{- 2 - 3 \cdot x + 3 + (x - 1) \cdot 3}$

Apply the distributive property.

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 x}{- 2 - 3 x + 3 + x \cdot 3 - 1 \cdot 3}$

Move $3$ to the left of $x$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 x}{- 2 - 3 x + 3 + 3 \cdot x - 1 \cdot 3}$

Multiply $- 1$ by $3$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 x}{- 2 - 3 x + 3 + 3 \cdot x - 3}$

Add $- 2$ and $3$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 x}{- 3 x + 1 + 3 x - 3}$

Add $- 3 x$ and $3 x$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 x}{0 + 1 - 3}$

Add $0$ and $1$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 x}{1 - 3}$

Subtract $3$ from $1$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 x}{- 2}$

Cancel the common factor of $- 2$ .

Cancel the common factor.

$h^{- 1} (- \frac{2}{x - 1} - 3) = \frac{- 2 x}{- 2}$

Divide $x$ by $1$ .

$h^{- 1} (- \frac{2}{x - 1} - 3) = x$

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

Set up the composite result function.

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

Evaluate $h (\frac{x + 1}{x + 3})$ by substituting in the value of $h^{- 1}$ into $h$ .

$h (\frac{x + 1}{x + 3}) = - \frac{2}{(\frac{x + 1}{x + 3}) - 1} - 3$

Simplify each term.

Simplify the denominator.

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$h (\frac{x + 1}{x + 3}) = - \frac{2}{\frac{x + 1}{x + 3} - 1 \cdot \frac{x + 3}{x + 3}} - 3$

Combine $- 1$ and $\frac{x + 3}{x + 3}$ .

$h (\frac{x + 1}{x + 3}) = - \frac{2}{\frac{x + 1}{x + 3} + \frac{- (x + 3)}{x + 3}} - 3$

Combine the numerators over the common denominator.

$h (\frac{x + 1}{x + 3}) = - \frac{2}{\frac{x + 1 - (x + 3)}{x + 3}} - 3$

Rewrite $\frac{x + 1 - (x + 3)}{x + 3}$ in a factored form.

Apply the distributive property.

$h (\frac{x + 1}{x + 3}) = - \frac{2}{\frac{x + 1 - x - 1 \cdot 3}{x + 3}} - 3$

Multiply $- 1$ by $3$ .

$h (\frac{x + 1}{x + 3}) = - \frac{2}{\frac{x + 1 - x - 3}{x + 3}} - 3$

Subtract $x$ from $x$ .

$h (\frac{x + 1}{x + 3}) = - \frac{2}{\frac{0 + 1 - 3}{x + 3}} - 3$

Add $0$ and $1$ .

$h (\frac{x + 1}{x + 3}) = - \frac{2}{\frac{1 - 3}{x + 3}} - 3$

Subtract $3$ from $1$ .

$h (\frac{x + 1}{x + 3}) = - \frac{2}{\frac{- 2}{x + 3}} - 3$

Move the negative in front of the fraction.

$h (\frac{x + 1}{x + 3}) = - \frac{2}{- \frac{2}{x + 3}} - 3$

Multiply the numerator by the reciprocal of the denominator.

$h (\frac{x + 1}{x + 3}) = - (2 (- \frac{x + 3}{2})) - 3$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{x + 3}{2}$ into the numerator.

$h (\frac{x + 1}{x + 3}) = - (2 (\frac{- (x + 3)}{2})) - 3$

Cancel the common factor.

$h (\frac{x + 1}{x + 3}) = - (2 (\frac{- (x + 3)}{2})) - 3$

Rewrite the expression.

$h (\frac{x + 1}{x + 3}) = (x + 3) - 3$

Apply the distributive property.

$h (\frac{x + 1}{x + 3}) = - (- x - 1 \cdot 3) - 3$

Multiply $- 1$ by $3$ .

$h (\frac{x + 1}{x + 3}) = - (- x - 3) - 3$

Apply the distributive property.

$h (\frac{x + 1}{x + 3}) = x + 3 - 3$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$h (\frac{x + 1}{x + 3}) = 1 x + 3 - 3$

Multiply $x$ by $1$ .

$h (\frac{x + 1}{x + 3}) = x + 3 - 3$

Multiply $- 1$ by $- 3$ .

$h (\frac{x + 1}{x + 3}) = x + 3 - 3$

Combine the opposite terms in $x + 3 - 3$ .

Subtract $3$ from $3$ .

$h (\frac{x + 1}{x + 3}) = x + 0$

Add $x$ and $0$ .

$h (\frac{x + 1}{x + 3}) = x$

Since $h^{- 1} (h (x)) = x$ and $h (h^{- 1} (x)) = x$ , then $h^{- 1} (x) = \frac{x + 1}{x + 3}$ is the inverse of $h (x) = - \frac{2}{x - 1} - 3$ .

$h^{- 1} (x) = \frac{x + 1}{x + 3}$

Find the Inverse h(x)=-2/(x-1)-3

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