Find the Inverse (x+9)/(x-4)

$\frac{x + 9}{x - 4}$

Step 1

Interchange the variables.

$x = \frac{y + 9}{y - 4}$

Step 2

Solve for $y$ .

Multiply the equation by $y - 4$ .

$(y - 4) (x) = (\frac{y + 9}{y - 4}) (y - 4)$

Simplify the left side.

Apply the distributive property.

$y x - 4 x = (\frac{y + 9}{y - 4}) (y - 4)$

Simplify the right side.

Cancel the common factor of $y - 4$ .

Cancel the common factor.

$y x - 4 x = \frac{y + 9}{y - 4} (y - 4)$

Rewrite the expression.

$y x - 4 x = y + 9$

Solve for $y$ .

Subtract $y$ from both sides of the equation.

$y x - 4 x - y = 9$

Add $4 x$ to both sides of the equation.

$y x - y = 9 + 4 x$

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

Factor $y$ out of $y x$ .

$y (x) - y = 9 + 4 x$

Factor $y$ out of $- y$ .

$y (x) + y \cdot - 1 = 9 + 4 x$

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

$y (x - 1) = 9 + 4 x$

Divide each term in $y (x - 1) = 9 + 4 x$ by $x - 1$ and simplify.

Divide each term in $y (x - 1) = 9 + 4 x$ by $x - 1$ .

$\frac{y (x - 1)}{x - 1} = \frac{9}{x - 1} + \frac{4 x}{x - 1}$

Simplify the left side.

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$\frac{y (x - 1)}{x - 1} = \frac{9}{x - 1} + \frac{4 x}{x - 1}$

Divide $y$ by $1$ .

$y = \frac{9}{x - 1} + \frac{4 x}{x - 1}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{9 + 4 x}{x - 1}$

Step 3

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

$f^{- 1} (x) = \frac{9 + 4 x}{x - 1}$

Step 4

Verify if $f^{- 1} (x) = \frac{9 + 4 x}{x - 1}$ is the inverse of $f (x) = \frac{x + 9}{x - 4}$ .

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 $f^{- 1} (\frac{x + 9}{x - 4})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{9 + 4 (\frac{x + 9}{x - 4})}{(\frac{x + 9}{x - 4}) - 1}$

Multiply the numerator and denominator of the complex fraction by $x - 4$ .

Multiply $\frac{9 + 4 \frac{x + 9}{x - 4}}{\frac{x + 9}{x - 4} - 1}$ by $\frac{x - 4}{x - 4}$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{x - 4}{x - 4} \cdot \frac{9 + 4 (\frac{x + 9}{x - 4})}{\frac{x + 9}{x - 4} - 1}$

Combine.

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (9 + 4 (\frac{x + 9}{x - 4}))}{(x - 4) (\frac{x + 9}{x - 4} - 1)}$

Apply the distributive property.

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) \cdot 9 + (x - 4) (4 (\frac{x + 9}{x - 4}))}{(x - 4) (\frac{x + 9}{x - 4}) + (x - 4) \cdot - 1}$

Cancel the common factor of $x - 4$ .

Cancel the common factor.

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) \cdot 9 + (x - 4) (4 (\frac{x + 9}{x - 4}))}{(x - 4) (\frac{x + 9}{x - 4}) + (x - 4) \cdot - 1}$

Rewrite the expression.

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) \cdot 9 + (x - 4) (4 (\frac{x + 9}{x - 4}))}{x + 9 + (x - 4) \cdot - 1}$

Simplify the numerator.

Factor $x - 4$ out of $(x - 4) \cdot 9 + (x - 4) \cdot 4 \frac{x + 9}{x - 4}$ .

Factor $x - 4$ out of $(x - 4) \cdot 9$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (9) + (x - 4) (4 (\frac{x + 9}{x - 4}))}{x + 9 + (x - 4) \cdot - 1}$

Factor $x - 4$ out of $(x - 4) (9) + (x - 4) (4 \frac{x + 9}{x - 4})$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (9 + 4 (\frac{x + 9}{x - 4}))}{x + 9 + (x - 4) \cdot - 1}$

Combine $4$ and $\frac{x + 9}{x - 4}$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (9 + \frac{4 (x + 9)}{x - 4})}{x + 9 + (x - 4) \cdot - 1}$

To write $9$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (\frac{9 (x - 4)}{x - 4} + \frac{4 (x + 9)}{x - 4})}{x + 9 + (x - 4) \cdot - 1}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (\frac{9 (x - 4) + 4 (x + 9)}{x - 4})}{x + 9 + (x - 4) \cdot - 1}$

Rewrite $\frac{9 (x - 4) + 4 (x + 9)}{x - 4}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (\frac{9 x + 9 \cdot - 4 + 4 (x + 9)}{x - 4})}{x + 9 + (x - 4) \cdot - 1}$

Multiply $9$ by $- 4$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (\frac{9 x - 36 + 4 (x + 9)}{x - 4})}{x + 9 + (x - 4) \cdot - 1}$

Apply the distributive property.

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (\frac{9 x - 36 + 4 x + 4 \cdot 9}{x - 4})}{x + 9 + (x - 4) \cdot - 1}$

Multiply $4$ by $9$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (\frac{9 x - 36 + 4 x + 36}{x - 4})}{x + 9 + (x - 4) \cdot - 1}$

Add $9 x$ and $4 x$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (\frac{13 x - 36 + 36}{x - 4})}{x + 9 + (x - 4) \cdot - 1}$

Add $- 36$ and $36$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (\frac{13 x + 0}{x - 4})}{x + 9 + (x - 4) \cdot - 1}$

Add $13 x$ and $0$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (\frac{13 x}{x - 4})}{x + 9 + (x - 4) \cdot - 1}$

Simplify the denominator.

Apply the distributive property.

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (\frac{13 x}{x - 4})}{x + 9 + x \cdot - 1 - 4 \cdot - 1}$

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

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (\frac{13 x}{x - 4})}{x + 9 - 1 \cdot x - 4 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (\frac{13 x}{x - 4})}{x + 9 - 1 \cdot x + 4}$

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

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (\frac{13 x}{x - 4})}{x + 9 - x + 4}$

Subtract $x$ from $x$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (\frac{13 x}{x - 4})}{0 + 9 + 4}$

Add $0$ and $9$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (\frac{13 x}{x - 4})}{9 + 4}$

Add $9$ and $4$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{(x - 4) (\frac{13 x}{x - 4})}{13}$

Reduce the expression by cancelling the common factors.

Factor $\frac{13 x}{x - 4}$ out of $\frac{(x - 4) \frac{13 x}{x - 4}}{13}$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{13 x}{x - 4} \cdot \frac{x - 4}{13}$

Cancel the common factor of $13$ .

Factor $13$ out of $13 x$ .

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{13 (x)}{x - 4} \cdot \frac{x - 4}{13}$

Cancel the common factor.

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{13 x}{x - 4} \cdot \frac{x - 4}{13}$

Rewrite the expression.

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{x}{x - 4} \cdot (x - 4)$

Cancel the common factor of $x - 4$ .

Cancel the common factor.

$f^{- 1} (\frac{x + 9}{x - 4}) = \frac{x}{x - 4} \cdot (x - 4)$

Rewrite the expression.

$f^{- 1} (\frac{x + 9}{x - 4}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{9 + 4 x}{x - 1})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{9 + 4 x}{x - 1}) = \frac{(\frac{9 + 4 x}{x - 1}) + 9}{(\frac{9 + 4 x}{x - 1}) - 4}$

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

Multiply $\frac{\frac{9 + 4 x}{x - 1} + 9}{\frac{9 + 4 x}{x - 1} - 4}$ by $\frac{x - 1}{x - 1}$ .

$f (\frac{9 + 4 x}{x - 1}) = \frac{x - 1}{x - 1} \cdot \frac{\frac{9 + 4 x}{x - 1} + 9}{\frac{9 + 4 x}{x - 1} - 4}$

Combine.

$f (\frac{9 + 4 x}{x - 1}) = \frac{(x - 1) (\frac{9 + 4 x}{x - 1} + 9)}{(x - 1) (\frac{9 + 4 x}{x - 1} - 4)}$

Apply the distributive property.

$f (\frac{9 + 4 x}{x - 1}) = \frac{(x - 1) (\frac{9 + 4 x}{x - 1}) + (x - 1) \cdot 9}{(x - 1) (\frac{9 + 4 x}{x - 1}) + (x - 1) \cdot - 4}$

Simplify by cancelling.

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$f (\frac{9 + 4 x}{x - 1}) = \frac{(x - 1) (\frac{9 + 4 x}{x - 1}) + (x - 1) \cdot 9}{(x - 1) (\frac{9 + 4 x}{x - 1}) + (x - 1) \cdot - 4}$

Rewrite the expression.

$f (\frac{9 + 4 x}{x - 1}) = \frac{9 + 4 x + (x - 1) \cdot 9}{(x - 1) (\frac{9 + 4 x}{x - 1}) + (x - 1) \cdot - 4}$

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$f (\frac{9 + 4 x}{x - 1}) = \frac{9 + 4 x + (x - 1) \cdot 9}{(x - 1) (\frac{9 + 4 x}{x - 1}) + (x - 1) \cdot - 4}$

Rewrite the expression.

$f (\frac{9 + 4 x}{x - 1}) = \frac{9 + 4 x + (x - 1) \cdot 9}{9 + 4 x + (x - 1) \cdot - 4}$

Simplify the numerator.

Apply the distributive property.

$f (\frac{9 + 4 x}{x - 1}) = \frac{9 + 4 x + x \cdot 9 - 1 \cdot 9}{9 + 4 x + (x - 1) \cdot - 4}$

Move $9$ to the left of $x$ .

$f (\frac{9 + 4 x}{x - 1}) = \frac{9 + 4 x + 9 \cdot x - 1 \cdot 9}{9 + 4 x + (x - 1) \cdot - 4}$

Multiply $- 1$ by $9$ .

$f (\frac{9 + 4 x}{x - 1}) = \frac{9 + 4 x + 9 \cdot x - 9}{9 + 4 x + (x - 1) \cdot - 4}$

Subtract $9$ from $9$ .

$f (\frac{9 + 4 x}{x - 1}) = \frac{4 x + 9 x + 0}{9 + 4 x + (x - 1) \cdot - 4}$

Add $4 x + 9 x$ and $0$ .

$f (\frac{9 + 4 x}{x - 1}) = \frac{4 x + 9 x}{9 + 4 x + (x - 1) \cdot - 4}$

Add $4 x$ and $9 x$ .

$f (\frac{9 + 4 x}{x - 1}) = \frac{13 x}{9 + 4 x + (x - 1) \cdot - 4}$

Simplify the denominator.

Apply the distributive property.

$f (\frac{9 + 4 x}{x - 1}) = \frac{13 x}{9 + 4 x + x \cdot - 4 - 1 \cdot - 4}$

Move $- 4$ to the left of $x$ .

$f (\frac{9 + 4 x}{x - 1}) = \frac{13 x}{9 + 4 x - 4 \cdot x - 1 \cdot - 4}$

Multiply $- 1$ by $- 4$ .

$f (\frac{9 + 4 x}{x - 1}) = \frac{13 x}{9 + 4 x - 4 \cdot x + 4}$

Add $9$ and $4$ .

$f (\frac{9 + 4 x}{x - 1}) = \frac{13 x}{4 x - 4 x + 13}$

Subtract $4 x$ from $4 x$ .

$f (\frac{9 + 4 x}{x - 1}) = \frac{13 x}{0 + 13}$

Add $0$ and $13$ .

$f (\frac{9 + 4 x}{x - 1}) = \frac{13 x}{13}$

Cancel the common factor of $13$ .

Cancel the common factor.

$f (\frac{9 + 4 x}{x - 1}) = \frac{13 x}{13}$

Divide $x$ by $1$ .

$f (\frac{9 + 4 x}{x - 1}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{9 + 4 x}{x - 1}$ is the inverse of $f (x) = \frac{x + 9}{x - 4}$ .

$f^{- 1} (x) = \frac{9 + 4 x}{x - 1}$

Find the Inverse (x+9)/(x-4)

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