Find the Inverse (9x+3)/(x-8)

$\frac{9 x + 3}{x - 8}$

Step 1

Interchange the variables.

$x = \frac{9 y + 3}{y - 8}$

Step 2

Solve for $y$ .

Multiply the equation by $y - 8$ .

$(y - 8) (x) = (\frac{9 y + 3}{y - 8}) (y - 8)$

Simplify the left side.

Apply the distributive property.

$y x - 8 x = (\frac{9 y + 3}{y - 8}) (y - 8)$

Simplify the right side.

Simplify $(\frac{9 y + 3}{y - 8}) (y - 8)$ .

Factor $3$ out of $9 y + 3$ .

Factor $3$ out of $9 y$ .

$y x - 8 x = \frac{3 (3 y) + 3}{y - 8} (y - 8)$

Factor $3$ out of $3$ .

$y x - 8 x = \frac{3 (3 y) + 3 (1)}{y - 8} (y - 8)$

Factor $3$ out of $3 (3 y) + 3 (1)$ .

$y x - 8 x = \frac{3 (3 y + 1)}{y - 8} (y - 8)$

Cancel the common factor of $y - 8$ .

Cancel the common factor.

$y x - 8 x = \frac{3 (3 y + 1)}{y - 8} (y - 8)$

Rewrite the expression.

$y x - 8 x = 3 (3 y + 1)$

Apply the distributive property.

$y x - 8 x = 3 (3 y) + 3 \cdot 1$

Multiply.

Multiply $3$ by $3$ .

$y x - 8 x = 9 y + 3 \cdot 1$

Multiply $3$ by $1$ .

$y x - 8 x = 9 y + 3$

Solve for $y$ .

Subtract $9 y$ from both sides of the equation.

$y x - 8 x - 9 y = 3$

Add $8 x$ to both sides of the equation.

$y x - 9 y = 3 + 8 x$

Factor $y$ out of $y x - 9 y$ .

Factor $y$ out of $y x$ .

$y (x) - 9 y = 3 + 8 x$

Factor $y$ out of $- 9 y$ .

$y (x) + y \cdot - 9 = 3 + 8 x$

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

$y (x - 9) = 3 + 8 x$

Divide each term in $y (x - 9) = 3 + 8 x$ by $x - 9$ and simplify.

Divide each term in $y (x - 9) = 3 + 8 x$ by $x - 9$ .

$\frac{y (x - 9)}{x - 9} = \frac{3}{x - 9} + \frac{8 x}{x - 9}$

Simplify the left side.

Cancel the common factor of $x - 9$ .

Cancel the common factor.

$\frac{y (x - 9)}{x - 9} = \frac{3}{x - 9} + \frac{8 x}{x - 9}$

Divide $y$ by $1$ .

$y = \frac{3}{x - 9} + \frac{8 x}{x - 9}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{3 + 8 x}{x - 9}$

Step 3

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

$f^{- 1} (x) = \frac{3 + 8 x}{x - 9}$

Step 4

Verify if $f^{- 1} (x) = \frac{3 + 8 x}{x - 9}$ is the inverse of $f (x) = \frac{9 x + 3}{x - 8}$ .

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 $f^{- 1} (\frac{9 x + 3}{x - 8})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{3 + 8 (\frac{9 x + 3}{x - 8})}{(\frac{9 x + 3}{x - 8}) - 9}$

Multiply the numerator and denominator of the complex fraction by $x - 8$ .

Multiply $\frac{3 + 8 \frac{9 x + 3}{x - 8}}{\frac{9 x + 3}{x - 8} - 9}$ by $\frac{x - 8}{x - 8}$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{x - 8}{x - 8} \cdot \frac{3 + 8 (\frac{9 x + 3}{x - 8})}{\frac{9 x + 3}{x - 8} - 9}$

Combine.

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (3 + 8 (\frac{9 x + 3}{x - 8}))}{(x - 8) (\frac{9 x + 3}{x - 8} - 9)}$

Apply the distributive property.

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) \cdot 3 + (x - 8) (8 (\frac{9 x + 3}{x - 8}))}{(x - 8) (\frac{9 x + 3}{x - 8}) + (x - 8) \cdot - 9}$

Simplify with factoring out.

Factor $3$ out of $9 x + 3$ .

Factor $3$ out of $9 x$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) \cdot 3 + (x - 8) (8 (\frac{9 x + 3}{x - 8}))}{(x - 8) (\frac{3 (3 x) + 3}{x - 8}) + (x - 8) \cdot - 9}$

Factor $3$ out of $3$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) \cdot 3 + (x - 8) (8 (\frac{9 x + 3}{x - 8}))}{(x - 8) (\frac{3 (3 x) + 3 (1)}{x - 8}) + (x - 8) \cdot - 9}$

Factor $3$ out of $3 (3 x) + 3 (1)$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) \cdot 3 + (x - 8) (8 (\frac{9 x + 3}{x - 8}))}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Factor $3$ out of $9 x + 3$ .

Factor $3$ out of $9 x$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) \cdot 3 + (x - 8) \cdot (8 (\frac{3 (3 x) + 3}{x - 8}))}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Factor $3$ out of $3$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) \cdot 3 + (x - 8) \cdot (8 (\frac{3 (3 x) + 3 (1)}{x - 8}))}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Factor $3$ out of $3 (3 x) + 3 (1)$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) \cdot 3 + (x - 8) \cdot (8 (\frac{3 (3 x + 1)}{x - 8}))}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Simplify the numerator.

Factor $x - 8$ out of $(x - 8) \cdot 3 + (x - 8) \cdot 8 \frac{3 (3 x + 1)}{x - 8}$ .

Factor $x - 8$ out of $(x - 8) \cdot 3$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (3) + (x - 8) (8 (\frac{3 (3 x + 1)}{x - 8}))}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Factor $x - 8$ out of $(x - 8) (3) + (x - 8) (8 \frac{3 (3 x + 1)}{x - 8})$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (3 + 8 (\frac{3 (3 x + 1)}{x - 8}))}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Multiply $8 \frac{3 (3 x + 1)}{x - 8}$ .

Combine $8$ and $\frac{3 (3 x + 1)}{x - 8}$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (3 + \frac{8 (3 (3 x + 1))}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Multiply $3$ by $8$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (3 + \frac{24 (3 x + 1)}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{x - 8}{x - 8}$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{3 (x - 8)}{x - 8} + \frac{24 (3 x + 1)}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{3 (x - 8) + 24 (3 x + 1)}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Reorder terms.

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{24 (3 x + 1) + 3 (x - 8)}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Rewrite $\frac{24 (3 x + 1) + 3 (x - 8)}{x - 8}$ in a factored form.

Factor $3$ out of $24 (3 x + 1) + 3 (x - 8)$ .

Factor $3$ out of $24 (3 x + 1)$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{3 (8 (3 x + 1)) + 3 (x - 8)}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Factor $3$ out of $3 (8 (3 x + 1)) + 3 (x - 8)$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{3 (8 (3 x + 1) + x - 8)}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Apply the distributive property.

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{3 (8 (3 x) + 8 \cdot 1 + x - 8)}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Multiply $3$ by $8$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{3 (24 x + 8 \cdot 1 + x - 8)}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Multiply $8$ by $1$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{3 (24 x + 8 + x - 8)}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Add $24 x$ and $x$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{3 (25 x + 8 - 8)}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Subtract $8$ from $8$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{3 (25 x + 0)}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Add $25 x$ and $0$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{3 \cdot (25 x)}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Multiply $3$ by $25$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{75 x}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Simplify the denominator.

Factor $x - 8$ out of $(x - 8) \frac{3 (3 x + 1)}{x - 8} + (x - 8) \cdot - 9$ .

Factor $x - 8$ out of $(x - 8) \frac{3 (3 x + 1)}{x - 8}$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{75 x}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) \cdot - 9}$

Factor $x - 8$ out of $(x - 8) \cdot - 9$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{75 x}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) (- 9)}$

Factor $x - 8$ out of $(x - 8) (\frac{3 (3 x + 1)}{x - 8}) + (x - 8) (- 9)$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{75 x}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8} - 9)}$

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{x - 8}{x - 8}$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{75 x}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8} - 9 \cdot \frac{x - 8}{x - 8})}$

Combine $- 9$ and $\frac{x - 8}{x - 8}$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{75 x}{x - 8})}{(x - 8) (\frac{3 (3 x + 1)}{x - 8} + \frac{- 9 (x - 8)}{x - 8})}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{75 x}{x - 8})}{(x - 8) (\frac{3 (3 x + 1) - 9 (x - 8)}{x - 8})}$

Rewrite $\frac{3 (3 x + 1) - 9 (x - 8)}{x - 8}$ in a factored form.

Factor $3$ out of $3 (3 x + 1) - 9 (x - 8)$ .

Factor $3$ out of $- 9 (x - 8)$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{75 x}{x - 8})}{(x - 8) (\frac{3 (3 x + 1) + 3 (- 3 (x - 8))}{x - 8})}$

Factor $3$ out of $3 (3 x + 1) + 3 (- 3 (x - 8))$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{75 x}{x - 8})}{(x - 8) (\frac{3 (3 x + 1 - 3 (x - 8))}{x - 8})}$

Apply the distributive property.

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{75 x}{x - 8})}{(x - 8) (\frac{3 (3 x + 1 - 3 x - 3 \cdot - 8)}{x - 8})}$

Multiply $- 3$ by $- 8$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{75 x}{x - 8})}{(x - 8) (\frac{3 (3 x + 1 - 3 x + 24)}{x - 8})}$

Subtract $3 x$ from $3 x$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{75 x}{x - 8})}{(x - 8) (\frac{3 (0 + 1 + 24)}{x - 8})}$

Add $0$ and $1$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{75 x}{x - 8})}{(x - 8) (\frac{3 (1 + 24)}{x - 8})}$

Add $1$ and $24$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{75 x}{x - 8})}{(x - 8) (\frac{3 \cdot 25}{x - 8})}$

Multiply $3$ by $25$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{75 x}{x - 8})}{(x - 8) (\frac{75}{x - 8})}$

Cancel the common factor of $x - 8$ .

Cancel the common factor.

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{(x - 8) (\frac{75 x}{x - 8})}{(x - 8) (\frac{75}{x - 8})}$

Rewrite the expression.

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{\frac{75 x}{x - 8}}{\frac{75}{x - 8}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{75 x}{x - 8} \cdot \frac{x - 8}{75}$

Cancel the common factor of $75$ .

Factor $75$ out of $75 x$ .

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{75 (x)}{x - 8} \cdot \frac{x - 8}{75}$

Cancel the common factor.

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{75 x}{x - 8} \cdot \frac{x - 8}{75}$

Rewrite the expression.

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{x}{x - 8} \cdot (x - 8)$

Cancel the common factor of $x - 8$ .

Cancel the common factor.

$f^{- 1} (\frac{9 x + 3}{x - 8}) = \frac{x}{x - 8} \cdot (x - 8)$

Rewrite the expression.

$f^{- 1} (\frac{9 x + 3}{x - 8}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{3 + 8 x}{x - 9})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{9 (\frac{3 + 8 x}{x - 9}) + 3}{(\frac{3 + 8 x}{x - 9}) - 8}$

Multiply the numerator and denominator of the complex fraction by $x - 9$ .

Multiply $\frac{9 \frac{3 + 8 x}{x - 9} + 3}{\frac{3 + 8 x}{x - 9} - 8}$ by $\frac{x - 9}{x - 9}$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{x - 9}{x - 9} \cdot \frac{9 (\frac{3 + 8 x}{x - 9}) + 3}{\frac{3 + 8 x}{x - 9} - 8}$

Combine.

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) (9 (\frac{3 + 8 x}{x - 9}) + 3)}{(x - 9) (\frac{3 + 8 x}{x - 9} - 8)}$

Apply the distributive property.

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) (9 (\frac{3 + 8 x}{x - 9})) + (x - 9) \cdot 3}{(x - 9) (\frac{3 + 8 x}{x - 9}) + (x - 9) \cdot - 8}$

Cancel the common factor of $x - 9$ .

Cancel the common factor.

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) (9 (\frac{3 + 8 x}{x - 9})) + (x - 9) \cdot 3}{(x - 9) (\frac{3 + 8 x}{x - 9}) + (x - 9) \cdot - 8}$

Rewrite the expression.

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) (9 (\frac{3 + 8 x}{x - 9})) + (x - 9) \cdot 3}{3 + 8 x + (x - 9) \cdot - 8}$

Simplify the numerator.

Factor $(x - 9) \cdot 3$ out of $(x - 9) \cdot 9 \frac{3 + 8 x}{x - 9} + (x - 9) \cdot 3$ .

Factor $(x - 9) \cdot 3$ out of $(x - 9) \cdot 9 \frac{3 + 8 x}{x - 9}$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot (3 (3 (\frac{3 + 8 x}{x - 9}))) + (x - 9) \cdot 3}{3 + 8 x + (x - 9) \cdot - 8}$

Factor $(x - 9) \cdot 3$ out of $(x - 9) \cdot 3$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot (3 (3 (\frac{3 + 8 x}{x - 9}))) + (x - 9) \cdot (3 (1))}{3 + 8 x + (x - 9) \cdot - 8}$

Factor $(x - 9) \cdot 3$ out of $(x - 9) \cdot 3 (3 \frac{3 + 8 x}{x - 9}) + (x - 9) \cdot 3 (1)$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot (3 (3 (\frac{3 + 8 x}{x - 9}) + 1))}{3 + 8 x + (x - 9) \cdot - 8}$

Combine $3$ and $\frac{3 + 8 x}{x - 9}$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot (3 (\frac{3 (3 + 8 x)}{x - 9} + 1))}{3 + 8 x + (x - 9) \cdot - 8}$

Write $1$ as a fraction with a common denominator.

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot (3 (\frac{3 (3 + 8 x)}{x - 9} + \frac{x - 9}{x - 9}))}{3 + 8 x + (x - 9) \cdot - 8}$

Combine the numerators over the common denominator.

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot (3 (\frac{3 (3 + 8 x) + x - 9}{x - 9}))}{3 + 8 x + (x - 9) \cdot - 8}$

Simplify the numerator.

Apply the distributive property.

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot (3 (\frac{3 \cdot 3 + 3 (8 x) + x - 9}{x - 9}))}{3 + 8 x + (x - 9) \cdot - 8}$

Multiply $3$ by $3$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot (3 (\frac{9 + 3 (8 x) + x - 9}{x - 9}))}{3 + 8 x + (x - 9) \cdot - 8}$

Multiply $8$ by $3$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot (3 (\frac{9 + 24 x + x - 9}{x - 9}))}{3 + 8 x + (x - 9) \cdot - 8}$

Subtract $9$ from $9$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot (3 (\frac{24 x + x + 0}{x - 9}))}{3 + 8 x + (x - 9) \cdot - 8}$

Add $24 x + x$ and $0$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot (3 (\frac{24 x + x}{x - 9}))}{3 + 8 x + (x - 9) \cdot - 8}$

Add $24 x$ and $x$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot (3 (\frac{25 x}{x - 9}))}{3 + 8 x + (x - 9) \cdot - 8}$

Combine exponents.

Combine $\frac{25 x}{x - 9}$ and $3$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot \frac{25 x \cdot 3}{x - 9}}{3 + 8 x + (x - 9) \cdot - 8}$

Multiply $3$ by $25$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot \frac{75 x}{x - 9}}{3 + 8 x + (x - 9) \cdot - 8}$

Simplify the denominator.

Apply the distributive property.

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot \frac{75 x}{x - 9}}{3 + 8 x + x \cdot - 8 - 9 \cdot - 8}$

Move $- 8$ to the left of $x$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot \frac{75 x}{x - 9}}{3 + 8 x - 8 \cdot x - 9 \cdot - 8}$

Multiply $- 9$ by $- 8$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot \frac{75 x}{x - 9}}{3 + 8 x - 8 \cdot x + 72}$

Add $3$ and $72$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot \frac{75 x}{x - 9}}{8 x - 8 x + 75}$

Subtract $8 x$ from $8 x$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot \frac{75 x}{x - 9}}{0 + 75}$

Add $0$ and $75$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{(x - 9) \cdot \frac{75 x}{x - 9}}{75}$

Reduce the expression by cancelling the common factors.

Factor $\frac{75 x}{x - 9}$ out of $\frac{(x - 9) \cdot \frac{75 x}{x - 9}}{75}$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{75 x}{x - 9} \cdot \frac{x - 9}{75}$

Cancel the common factor of $75$ .

Factor $75$ out of $75 x$ .

$f (\frac{3 + 8 x}{x - 9}) = \frac{75 (x)}{x - 9} \cdot \frac{x - 9}{75}$

Cancel the common factor.

$f (\frac{3 + 8 x}{x - 9}) = \frac{75 x}{x - 9} \cdot \frac{x - 9}{75}$

Rewrite the expression.

$f (\frac{3 + 8 x}{x - 9}) = \frac{x}{x - 9} \cdot (x - 9)$

Cancel the common factor of $x - 9$ .

Cancel the common factor.

$f (\frac{3 + 8 x}{x - 9}) = \frac{x}{x - 9} \cdot (x - 9)$

Rewrite the expression.

$f (\frac{3 + 8 x}{x - 9}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{3 + 8 x}{x - 9}$ is the inverse of $f (x) = \frac{9 x + 3}{x - 8}$ .

$f^{- 1} (x) = \frac{3 + 8 x}{x - 9}$

Find the Inverse (9x+3)/(x-8)

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