Find the Inverse (x+8)/(x-5)

$\frac{x + 8}{x - 5}$

Interchange the variables.

$x = \frac{y + 8}{y - 5}$

Solve for $y$ .

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Multiply the equation by $y - 5$ .

$(y - 5) (x) = (\frac{y + 8}{y - 5}) (y - 5)$

Apply the distributive property.

$y x - 5 x = (\frac{y + 8}{y - 5}) (y - 5)$

Cancel the common factor of $y - 5$ .

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Cancel the common factor.

$y x - 5 x = \frac{y + 8}{y - 5} (y - 5)$

Rewrite the expression.

$y x - 5 x = y + 8$

Solve for $y$ .

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Subtract $y$ from both sides of the equation.

$y x - 5 x - y = 8$

Add $5 x$ to both sides of the equation.

$y x - y = 8 + 5 x$

Factor $y$ out of $y x - y$ .

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Factor $y$ out of $y x$ .

$y (x) - y = 8 + 5 x$

Factor $y$ out of $- y$ .

$y (x) + y \cdot - 1 = 8 + 5 x$

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

$y (x - 1) = 8 + 5 x$

Divide each term by $x - 1$ and simplify.

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Divide each term in $y (x - 1) = 8 + 5 x$ by $x - 1$ .

$\frac{y (x - 1)}{x - 1} = \frac{8}{x - 1} + \frac{5 x}{x - 1}$

Cancel the common factor of $x - 1$ .

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Cancel the common factor.

$\frac{y (x - 1)}{x - 1} = \frac{8}{x - 1} + \frac{5 x}{x - 1}$

Divide $y$ by $1$ .

$y = \frac{8}{x - 1} + \frac{5 x}{x - 1}$

Combine the numerators over the common denominator.

$y = \frac{8 + 5 x}{x - 1}$

Solve for $y$ and replace with $f^{- 1} (x)$ .

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

$f^{- 1} (x) = \frac{8 + 5 x}{x - 1}$

Set up the composite result function.

$g (f (x))$

Evaluate $g (f (x))$ by substituting in the value of $f$ into $g$ .

$g (\frac{x + 8}{x - 5}) = \frac{8 + 5 (\frac{x + 8}{x - 5})}{(\frac{x + 8}{x - 5}) - 1}$

Multiply the numerator and denominator of the complex fraction by $x - 5$ .

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Multiply $\frac{8 + 5 (\frac{x + 8}{x - 5})}{(\frac{x + 8}{x - 5}) - 1}$ by $\frac{x - 5}{x - 5}$ .

$g (\frac{x + 8}{x - 5}) = \frac{x - 5}{x - 5} \cdot \frac{8 + 5 (\frac{x + 8}{x - 5})}{(\frac{x + 8}{x - 5}) - 1}$

Combine.

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (8 + 5 (\frac{x + 8}{x - 5}))}{(x - 5) ((\frac{x + 8}{x - 5}) - 1)}$

Apply the distributive property.

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) \cdot 8 + (x - 5) (5 (\frac{x + 8}{x - 5}))}{(x - 5) (\frac{x + 8}{x - 5}) + (x - 5) \cdot - 1}$

Cancel the common factor of $x - 5$ .

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Cancel the common factor.

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) \cdot 8 + (x - 5) (5 (\frac{x + 8}{x - 5}))}{(x - 5) (\frac{x + 8}{x - 5}) + (x - 5) \cdot - 1}$

Rewrite the expression.

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) \cdot 8 + (x - 5) (5 (\frac{x + 8}{x - 5}))}{x + 8 + (x - 5) \cdot - 1}$

Simplify the numerator.

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Factor $x - 5$ out of $(x - 5) \cdot 8 + (x - 5) (5 (\frac{x + 8}{x - 5}))$ .

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Factor $x - 5$ out of $(x - 5) \cdot 8$ .

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (8) + (x - 5) (5 (\frac{x + 8}{x - 5}))}{x + 8 + (x - 5) \cdot - 1}$

Factor $x - 5$ out of $(x - 5) (8) + (x - 5) (5 (\frac{x + 8}{x - 5}))$ .

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (8 + 5 (\frac{x + 8}{x - 5}))}{x + 8 + (x - 5) \cdot - 1}$

Combine $5$ and $\frac{x + 8}{x - 5}$ .

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (8 + \frac{5 (x + 8)}{x - 5})}{x + 8 + (x - 5) \cdot - 1}$

To write $8$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (\frac{8 (x - 5)}{x - 5} + \frac{5 (x + 8)}{x - 5})}{x + 8 + (x - 5) \cdot - 1}$

Combine the numerators over the common denominator.

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (\frac{8 (x - 5) + 5 (x + 8)}{x - 5})}{x + 8 + (x - 5) \cdot - 1}$

Rewrite $\frac{8 (x - 5) + 5 (x + 8)}{x - 5}$ in a factored form.

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Apply the distributive property.

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (\frac{8 x + 8 \cdot - 5 + 5 (x + 8)}{x - 5})}{x + 8 + (x - 5) \cdot - 1}$

Multiply $8$ by $- 5$ .

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (\frac{8 x - 40 + 5 (x + 8)}{x - 5})}{x + 8 + (x - 5) \cdot - 1}$

Apply the distributive property.

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (\frac{8 x - 40 + 5 x + 5 \cdot 8}{x - 5})}{x + 8 + (x - 5) \cdot - 1}$

Multiply $5$ by $8$ .

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (\frac{8 x - 40 + 5 x + 40}{x - 5})}{x + 8 + (x - 5) \cdot - 1}$

Add $8 x$ and $5 x$ .

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (\frac{13 x - 40 + 40}{x - 5})}{x + 8 + (x - 5) \cdot - 1}$

Add $- 40$ and $40$ .

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (\frac{13 x + 0}{x - 5})}{x + 8 + (x - 5) \cdot - 1}$

Add $13 x$ and $0$ .

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (\frac{13 x}{x - 5})}{x + 8 + (x - 5) \cdot - 1}$

Simplify the denominator.

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Apply the distributive property.

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (\frac{13 x}{x - 5})}{x + 8 + x \cdot - 1 - 5 \cdot - 1}$

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

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (\frac{13 x}{x - 5})}{x + 8 - 1 \cdot x - 5 \cdot - 1}$

Multiply $- 5$ by $- 1$ .

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (\frac{13 x}{x - 5})}{x + 8 - 1 \cdot x + 5}$

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

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (\frac{13 x}{x - 5})}{x + 8 - x + 5}$

Subtract $x$ from $x$ .

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (\frac{13 x}{x - 5})}{0 + 8 + 5}$

Add $0$ and $8$ .

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (\frac{13 x}{x - 5})}{8 + 5}$

Add $8$ and $5$ .

$g (\frac{x + 8}{x - 5}) = \frac{(x - 5) (\frac{13 x}{x - 5})}{13}$

Reduce the expression by cancelling the common factors.

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Factor $\frac{13 x}{x - 5}$ out of $\frac{(x - 5) \frac{13 x}{x - 5}}{13}$ .

$g (\frac{x + 8}{x - 5}) = \frac{13 x}{x - 5} \cdot \frac{x - 5}{13}$

Cancel the common factor of $13$ .

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Factor $13$ out of $13 x$ .

$g (\frac{x + 8}{x - 5}) = \frac{13 (x)}{x - 5} \cdot \frac{x - 5}{13}$

Cancel the common factor.

$g (\frac{x + 8}{x - 5}) = \frac{13 x}{x - 5} \cdot \frac{x - 5}{13}$

Rewrite the expression.

$g (\frac{x + 8}{x - 5}) = \frac{x}{x - 5} \cdot (x - 5)$

Cancel the common factor of $x - 5$ .

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Cancel the common factor.

$g (\frac{x + 8}{x - 5}) = \frac{x}{x - 5} \cdot (x - 5)$

Rewrite the expression.

$g (\frac{x + 8}{x - 5}) = x$

Since $g (f (x)) = x$ , $f^{- 1} (x) = \frac{8 + 5 x}{x - 1}$ is the inverse of $f (x) = \frac{x + 8}{x - 5}$ .

$f^{- 1} (x) = \frac{8 + 5 x}{x - 1}$

Find the Inverse (x+8)/(x-5)

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