Find the Inverse (x+9)/(x-7)

$\frac{x + 9}{x - 7}$

Step 1

Interchange the variables.

$x = \frac{y + 9}{y - 7}$

Step 2

Solve for $y$ .

Multiply the equation by $y - 7$ .

$(y - 7) (x) = (\frac{y + 9}{y - 7}) (y - 7)$

Simplify the left side.

Apply the distributive property.

$y x - 7 x = (\frac{y + 9}{y - 7}) (y - 7)$

Simplify the right side.

Cancel the common factor of $y - 7$ .

Cancel the common factor.

$y x - 7 x = \frac{y + 9}{y - 7} (y - 7)$

Rewrite the expression.

$y x - 7 x = y + 9$

Solve for $y$ .

Subtract $y$ from both sides of the equation.

$y x - 7 x - y = 9$

Add $7 x$ to both sides of the equation.

$y x - y = 9 + 7 x$

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

Factor $y$ out of $y x$ .

$y (x) - y = 9 + 7 x$

Factor $y$ out of $- y$ .

$y (x) + y \cdot - 1 = 9 + 7 x$

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

$y (x - 1) = 9 + 7 x$

Divide each term in $y (x - 1) = 9 + 7 x$ by $x - 1$ and simplify.

Divide each term in $y (x - 1) = 9 + 7 x$ by $x - 1$ .

$\frac{y (x - 1)}{x - 1} = \frac{9}{x - 1} + \frac{7 x}{x - 1}$

Simplify the left side.

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$\frac{y (x - 1)}{x - 1} = \frac{9}{x - 1} + \frac{7 x}{x - 1}$

Divide $y$ by $1$ .

$y = \frac{9}{x - 1} + \frac{7 x}{x - 1}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{9 + 7 x}{x - 1}$

Step 3

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

$f^{- 1} (x) = \frac{9 + 7 x}{x - 1}$

Step 4

Verify if $f^{- 1} (x) = \frac{9 + 7 x}{x - 1}$ is the inverse of $f (x) = \frac{x + 9}{x - 7}$ .

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 $f^{- 1} (\frac{x + 9}{x - 7})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{9 + 7 (\frac{x + 9}{x - 7})}{(\frac{x + 9}{x - 7}) - 1}$

Multiply the numerator and denominator of the complex fraction by $x - 7$ .

Multiply $\frac{9 + 7 \frac{x + 9}{x - 7}}{\frac{x + 9}{x - 7} - 1}$ by $\frac{x - 7}{x - 7}$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{x - 7}{x - 7} \cdot \frac{9 + 7 (\frac{x + 9}{x - 7})}{\frac{x + 9}{x - 7} - 1}$

Combine.

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (9 + 7 (\frac{x + 9}{x - 7}))}{(x - 7) (\frac{x + 9}{x - 7} - 1)}$

Apply the distributive property.

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) \cdot 9 + (x - 7) (7 (\frac{x + 9}{x - 7}))}{(x - 7) (\frac{x + 9}{x - 7}) + (x - 7) \cdot - 1}$

Cancel the common factor of $x - 7$ .

Cancel the common factor.

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) \cdot 9 + (x - 7) (7 (\frac{x + 9}{x - 7}))}{(x - 7) (\frac{x + 9}{x - 7}) + (x - 7) \cdot - 1}$

Rewrite the expression.

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) \cdot 9 + (x - 7) (7 (\frac{x + 9}{x - 7}))}{x + 9 + (x - 7) \cdot - 1}$

Simplify the numerator.

Factor $x - 7$ out of $(x - 7) \cdot 9 + (x - 7) \cdot 7 \frac{x + 9}{x - 7}$ .

Factor $x - 7$ out of $(x - 7) \cdot 9$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (9) + (x - 7) (7 (\frac{x + 9}{x - 7}))}{x + 9 + (x - 7) \cdot - 1}$

Factor $x - 7$ out of $(x - 7) (9) + (x - 7) (7 \frac{x + 9}{x - 7})$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (9 + 7 (\frac{x + 9}{x - 7}))}{x + 9 + (x - 7) \cdot - 1}$

Combine $7$ and $\frac{x + 9}{x - 7}$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (9 + \frac{7 (x + 9)}{x - 7})}{x + 9 + (x - 7) \cdot - 1}$

To write $9$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (\frac{9 (x - 7)}{x - 7} + \frac{7 (x + 9)}{x - 7})}{x + 9 + (x - 7) \cdot - 1}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (\frac{9 (x - 7) + 7 (x + 9)}{x - 7})}{x + 9 + (x - 7) \cdot - 1}$

Rewrite $\frac{9 (x - 7) + 7 (x + 9)}{x - 7}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (\frac{9 x + 9 \cdot - 7 + 7 (x + 9)}{x - 7})}{x + 9 + (x - 7) \cdot - 1}$

Multiply $9$ by $- 7$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (\frac{9 x - 63 + 7 (x + 9)}{x - 7})}{x + 9 + (x - 7) \cdot - 1}$

Apply the distributive property.

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (\frac{9 x - 63 + 7 x + 7 \cdot 9}{x - 7})}{x + 9 + (x - 7) \cdot - 1}$

Multiply $7$ by $9$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (\frac{9 x - 63 + 7 x + 63}{x - 7})}{x + 9 + (x - 7) \cdot - 1}$

Add $9 x$ and $7 x$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (\frac{16 x - 63 + 63}{x - 7})}{x + 9 + (x - 7) \cdot - 1}$

Add $- 63$ and $63$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (\frac{16 x + 0}{x - 7})}{x + 9 + (x - 7) \cdot - 1}$

Add $16 x$ and $0$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (\frac{16 x}{x - 7})}{x + 9 + (x - 7) \cdot - 1}$

Simplify the denominator.

Apply the distributive property.

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (\frac{16 x}{x - 7})}{x + 9 + x \cdot - 1 - 7 \cdot - 1}$

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

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (\frac{16 x}{x - 7})}{x + 9 - 1 \cdot x - 7 \cdot - 1}$

Multiply $- 7$ by $- 1$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (\frac{16 x}{x - 7})}{x + 9 - 1 \cdot x + 7}$

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

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (\frac{16 x}{x - 7})}{x + 9 - x + 7}$

Subtract $x$ from $x$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (\frac{16 x}{x - 7})}{0 + 9 + 7}$

Add $0$ and $9$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (\frac{16 x}{x - 7})}{9 + 7}$

Add $9$ and $7$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{(x - 7) (\frac{16 x}{x - 7})}{16}$

Reduce the expression by cancelling the common factors.

Factor $\frac{16 x}{x - 7}$ out of $\frac{(x - 7) \frac{16 x}{x - 7}}{16}$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{16 x}{x - 7} \cdot \frac{x - 7}{16}$

Cancel the common factor of $16$ .

Factor $16$ out of $16 x$ .

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{16 (x)}{x - 7} \cdot \frac{x - 7}{16}$

Cancel the common factor.

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{16 x}{x - 7} \cdot \frac{x - 7}{16}$

Rewrite the expression.

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{x}{x - 7} \cdot (x - 7)$

Cancel the common factor of $x - 7$ .

Cancel the common factor.

$f^{- 1} (\frac{x + 9}{x - 7}) = \frac{x}{x - 7} \cdot (x - 7)$

Rewrite the expression.

$f^{- 1} (\frac{x + 9}{x - 7}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{9 + 7 x}{x - 1})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{9 + 7 x}{x - 1}) = \frac{(\frac{9 + 7 x}{x - 1}) + 9}{(\frac{9 + 7 x}{x - 1}) - 7}$

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

Multiply $\frac{\frac{9 + 7 x}{x - 1} + 9}{\frac{9 + 7 x}{x - 1} - 7}$ by $\frac{x - 1}{x - 1}$ .

$f (\frac{9 + 7 x}{x - 1}) = \frac{x - 1}{x - 1} \cdot \frac{\frac{9 + 7 x}{x - 1} + 9}{\frac{9 + 7 x}{x - 1} - 7}$

Combine.

$f (\frac{9 + 7 x}{x - 1}) = \frac{(x - 1) (\frac{9 + 7 x}{x - 1} + 9)}{(x - 1) (\frac{9 + 7 x}{x - 1} - 7)}$

Apply the distributive property.

$f (\frac{9 + 7 x}{x - 1}) = \frac{(x - 1) (\frac{9 + 7 x}{x - 1}) + (x - 1) \cdot 9}{(x - 1) (\frac{9 + 7 x}{x - 1}) + (x - 1) \cdot - 7}$

Simplify by cancelling.

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$f (\frac{9 + 7 x}{x - 1}) = \frac{(x - 1) (\frac{9 + 7 x}{x - 1}) + (x - 1) \cdot 9}{(x - 1) (\frac{9 + 7 x}{x - 1}) + (x - 1) \cdot - 7}$

Rewrite the expression.

$f (\frac{9 + 7 x}{x - 1}) = \frac{9 + 7 x + (x - 1) \cdot 9}{(x - 1) (\frac{9 + 7 x}{x - 1}) + (x - 1) \cdot - 7}$

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$f (\frac{9 + 7 x}{x - 1}) = \frac{9 + 7 x + (x - 1) \cdot 9}{(x - 1) (\frac{9 + 7 x}{x - 1}) + (x - 1) \cdot - 7}$

Rewrite the expression.

$f (\frac{9 + 7 x}{x - 1}) = \frac{9 + 7 x + (x - 1) \cdot 9}{9 + 7 x + (x - 1) \cdot - 7}$

Simplify the numerator.

Apply the distributive property.

$f (\frac{9 + 7 x}{x - 1}) = \frac{9 + 7 x + x \cdot 9 - 1 \cdot 9}{9 + 7 x + (x - 1) \cdot - 7}$

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

$f (\frac{9 + 7 x}{x - 1}) = \frac{9 + 7 x + 9 \cdot x - 1 \cdot 9}{9 + 7 x + (x - 1) \cdot - 7}$

Multiply $- 1$ by $9$ .

$f (\frac{9 + 7 x}{x - 1}) = \frac{9 + 7 x + 9 \cdot x - 9}{9 + 7 x + (x - 1) \cdot - 7}$

Subtract $9$ from $9$ .

$f (\frac{9 + 7 x}{x - 1}) = \frac{7 x + 9 x + 0}{9 + 7 x + (x - 1) \cdot - 7}$

Add $7 x + 9 x$ and $0$ .

$f (\frac{9 + 7 x}{x - 1}) = \frac{7 x + 9 x}{9 + 7 x + (x - 1) \cdot - 7}$

Add $7 x$ and $9 x$ .

$f (\frac{9 + 7 x}{x - 1}) = \frac{16 x}{9 + 7 x + (x - 1) \cdot - 7}$

Simplify the denominator.

Apply the distributive property.

$f (\frac{9 + 7 x}{x - 1}) = \frac{16 x}{9 + 7 x + x \cdot - 7 - 1 \cdot - 7}$

Move $- 7$ to the left of $x$ .

$f (\frac{9 + 7 x}{x - 1}) = \frac{16 x}{9 + 7 x - 7 \cdot x - 1 \cdot - 7}$

Multiply $- 1$ by $- 7$ .

$f (\frac{9 + 7 x}{x - 1}) = \frac{16 x}{9 + 7 x - 7 \cdot x + 7}$

Add $9$ and $7$ .

$f (\frac{9 + 7 x}{x - 1}) = \frac{16 x}{7 x - 7 x + 16}$

Subtract $7 x$ from $7 x$ .

$f (\frac{9 + 7 x}{x - 1}) = \frac{16 x}{0 + 16}$

Add $0$ and $16$ .

$f (\frac{9 + 7 x}{x - 1}) = \frac{16 x}{16}$

Cancel the common factor of $16$ .

Cancel the common factor.

$f (\frac{9 + 7 x}{x - 1}) = \frac{16 x}{16}$

Divide $x$ by $1$ .

$f (\frac{9 + 7 x}{x - 1}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{9 + 7 x}{x - 1}$ is the inverse of $f (x) = \frac{x + 9}{x - 7}$ .

$f^{- 1} (x) = \frac{9 + 7 x}{x - 1}$

Find the Inverse (x+9)/(x-7)

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