Find the Inverse (5x+8)/(-3x-1)

$\frac{5 x + 8}{- 3 x - 1}$

Step 1

Interchange the variables.

$x = \frac{5 y + 8}{- 3 y - 1}$

Step 2

Solve for $y$ .

Multiply the equation by $- 3 y - 1$ .

$(- 3 y - 1) (x) = (\frac{5 y + 8}{- 3 y - 1}) (- 3 y - 1)$

Simplify the left side.

Simplify $(- 3 y - 1) (x)$ .

Apply the distributive property.

$- 3 y x - 1 x = (\frac{5 y + 8}{- 3 y - 1}) (- 3 y - 1)$

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

$- 3 y x - x = (\frac{5 y + 8}{- 3 y - 1}) (- 3 y - 1)$

Simplify the right side.

Simplify $(\frac{5 y + 8}{- 3 y - 1}) (- 3 y - 1)$ .

Multiply $\frac{5 y + 8}{- 3 y - 1}$ by $- 3 y - 1$ .

$- 3 y x - x = \frac{(5 y + 8) (- 3 y - 1)}{- 3 y - 1}$

Cancel the common factor of $- 3 y - 1$ .

Cancel the common factor.

$- 3 y x - x = \frac{(5 y + 8) (- 3 y - 1)}{- 3 y - 1}$

Divide $5 y + 8$ by $1$ .

$- 3 y x - x = 5 y + 8$

Solve for $y$ .

Subtract $5 y$ from both sides of the equation.

$- 3 y x - x - 5 y = 8$

Add $x$ to both sides of the equation.

$- 3 y x - 5 y = 8 + x$

Factor $y$ out of $- 3 y x - 5 y$ .

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

$y (- 3 x) - 5 y = 8 + x$

Factor $y$ out of $- 5 y$ .

$y (- 3 x) + y \cdot - 5 = 8 + x$

Factor $y$ out of $y (- 3 x) + y \cdot - 5$ .

$y (- 3 x - 5) = 8 + x$

Divide each term in $y (- 3 x - 5) = 8 + x$ by $- 3 x - 5$ and simplify.

Divide each term in $y (- 3 x - 5) = 8 + x$ by $- 3 x - 5$ .

$\frac{y (- 3 x - 5)}{- 3 x - 5} = \frac{8}{- 3 x - 5} + \frac{x}{- 3 x - 5}$

Simplify the left side.

Cancel the common factor of $- 3 x - 5$ .

Cancel the common factor.

$\frac{y (- 3 x - 5)}{- 3 x - 5} = \frac{8}{- 3 x - 5} + \frac{x}{- 3 x - 5}$

Divide $y$ by $1$ .

$y = \frac{8}{- 3 x - 5} + \frac{x}{- 3 x - 5}$

Simplify the right side.

Combine the numerators over the common denominator.

$y = \frac{8 + x}{- 3 x - 5}$

Factor $- 1$ out of $- 3 x$ .

$y = \frac{8 + x}{- (3 x) - 5}$

Rewrite $- 5$ as $- 1 (5)$ .

$y = \frac{8 + x}{- (3 x) - 1 (5)}$

Factor $- 1$ out of $- (3 x) - 1 (5)$ .

$y = \frac{8 + x}{- (3 x + 5)}$

Rewrite negatives.

Rewrite $- (3 x + 5)$ as $- 1 (3 x + 5)$ .

$y = \frac{8 + x}{- 1 (3 x + 5)}$

Move the negative in front of the fraction.

$y = - \frac{8 + x}{3 x + 5}$

Step 3

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

$f^{- 1} (x) = - \frac{8 + x}{3 x + 5}$

Step 4

Verify if $f^{- 1} (x) = - \frac{8 + x}{3 x + 5}$ is the inverse of $f (x) = \frac{5 x + 8}{- 3 x - 1}$ .

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 $f^{- 1} (\frac{5 x + 8}{- 3 x - 1})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{8 + \frac{5 x + 8}{- 3 x - 1}}{3 (\frac{5 x + 8}{- 3 x - 1}) + 5}$

Remove parentheses.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{8 + \frac{5 x + 8}{- 3 x - 1}}{3 (\frac{5 x + 8}{- 3 x - 1}) + 5}$

Multiply the numerator and denominator of the complex fraction by $- 3 x - 1$ .

Multiply $\frac{8 + \frac{5 x + 8}{- 3 x - 1}}{3 \frac{5 x + 8}{- 3 x - 1} + 5}$ by $\frac{- 3 x - 1}{- 3 x - 1}$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{- 3 x - 1}{- 3 x - 1} \cdot \frac{8 + \frac{5 x + 8}{- 3 x - 1}}{3 (\frac{5 x + 8}{- 3 x - 1}) + 5})$

Combine.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{(- 3 x - 1) (8 + \frac{5 x + 8}{- 3 x - 1})}{(- 3 x - 1) (3 (\frac{5 x + 8}{- 3 x - 1}) + 5)}$

Apply the distributive property.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{(- 3 x - 1) \cdot 8 + (- 3 x - 1) (\frac{5 x + 8}{- 3 x - 1})}{(- 3 x - 1) (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) \cdot 5}$

Simplify the numerator.

Factor $- 3 x - 1$ out of $(- 3 x - 1) \cdot 8 + (- 3 x - 1) \frac{5 x + 8}{- 3 x - 1}$ .

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

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{(- 3 x - 1) (8) + (- 3 x - 1) (\frac{5 x + 8}{- 3 x - 1})}{(- 3 x - 1) \cdot (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) \cdot 5}$

Factor $- 3 x - 1$ out of $(- 3 x - 1) (8) + (- 3 x - 1) (\frac{5 x + 8}{- 3 x - 1})$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{(- 3 x - 1) (8 + \frac{5 x + 8}{- 3 x - 1})}{(- 3 x - 1) \cdot (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) \cdot 5}$

To write $8$ as a fraction with a common denominator, multiply by $\frac{- 3 x - 1}{- 3 x - 1}$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{(- 3 x - 1) (\frac{8 (- 3 x - 1)}{- 3 x - 1} + \frac{5 x + 8}{- 3 x - 1})}{(- 3 x - 1) \cdot (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) \cdot 5}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{(- 3 x - 1) (\frac{8 (- 3 x - 1) + 5 x + 8}{- 3 x - 1})}{(- 3 x - 1) \cdot (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) \cdot 5}$

Reorder terms.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{(- 3 x - 1) (\frac{5 x + 8 (- 3 x - 1) + 8}{- 3 x - 1})}{(- 3 x - 1) \cdot (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) \cdot 5}$

Rewrite $\frac{5 x + 8 (- 3 x - 1) + 8}{- 3 x - 1}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{(- 3 x - 1) (\frac{5 x + 8 (- 3 x) + 8 \cdot - 1 + 8}{- 3 x - 1})}{(- 3 x - 1) \cdot (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) \cdot 5}$

Multiply $- 3$ by $8$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{(- 3 x - 1) (\frac{5 x - 24 x + 8 \cdot - 1 + 8}{- 3 x - 1})}{(- 3 x - 1) \cdot (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) \cdot 5}$

Multiply $8$ by $- 1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{(- 3 x - 1) (\frac{5 x - 24 x - 8 + 8}{- 3 x - 1})}{(- 3 x - 1) \cdot (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) \cdot 5}$

Subtract $24 x$ from $5 x$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{(- 3 x - 1) (\frac{- 19 x - 8 + 8}{- 3 x - 1})}{(- 3 x - 1) \cdot (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) \cdot 5}$

Add $- 8$ and $8$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{(- 3 x - 1) (\frac{- 19 x + 0}{- 3 x - 1})}{(- 3 x - 1) \cdot (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) \cdot 5}$

Add $- 19 x$ and $0$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{(- 3 x - 1) (\frac{- 19 x}{- 3 x - 1})}{(- 3 x - 1) \cdot (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) \cdot 5}$

Move the negative in front of the fraction.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{(- 3 x - 1) (- \frac{(19) x}{- 3 x - 1})}{(- 3 x - 1) \cdot (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) \cdot 5}$

Remove unnecessary parentheses.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{(- 3 x - 1) \cdot (- 1 \frac{19 x}{- 3 x - 1})}{(- 3 x - 1) \cdot (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) \cdot 5}$

Factor out negative.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) \cdot (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) \cdot 5}$

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) \cdot 5}$

Simplify the denominator.

Factor $- 3 x - 1$ out of $(- 3 x - 1) \cdot 3 \frac{5 x + 8}{- 3 x - 1} + (- 3 x - 1) \cdot 5$ .

Factor $- 3 x - 1$ out of $(- 3 x - 1) \cdot 5$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (3 (\frac{5 x + 8}{- 3 x - 1})) + (- 3 x - 1) (5)}$

Factor $- 3 x - 1$ out of $(- 3 x - 1) (3 \frac{5 x + 8}{- 3 x - 1}) + (- 3 x - 1) (5)$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (3 (\frac{5 x + 8}{- 3 x - 1}) + 5)}$

Combine $3$ and $\frac{5 x + 8}{- 3 x - 1}$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (\frac{3 (5 x + 8)}{- 3 x - 1} + 5)}$

To write $5$ as a fraction with a common denominator, multiply by $\frac{- 3 x - 1}{- 3 x - 1}$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (\frac{3 (5 x + 8)}{- 3 x - 1} + \frac{5 (- 3 x - 1)}{- 3 x - 1})}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (\frac{3 (5 x + 8) + 5 (- 3 x - 1)}{- 3 x - 1})}$

Reorder terms.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (\frac{5 (- 3 x - 1) + 3 (5 x + 8)}{- 3 x - 1})}$

Rewrite $\frac{5 (- 3 x - 1) + 3 (5 x + 8)}{- 3 x - 1}$ in a factored form.

Apply the distributive property.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (\frac{5 (- 3 x) + 5 \cdot - 1 + 3 (5 x + 8)}{- 3 x - 1})}$

Multiply $- 3$ by $5$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (\frac{- 15 x + 5 \cdot - 1 + 3 (5 x + 8)}{- 3 x - 1})}$

Multiply $5$ by $- 1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (\frac{- 15 x - 5 + 3 (5 x + 8)}{- 3 x - 1})}$

Apply the distributive property.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (\frac{- 15 x - 5 + 3 (5 x) + 3 \cdot 8}{- 3 x - 1})}$

Multiply $5$ by $3$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (\frac{- 15 x - 5 + 15 x + 3 \cdot 8}{- 3 x - 1})}$

Multiply $3$ by $8$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (\frac{- 15 x - 5 + 15 x + 24}{- 3 x - 1})}$

Add $- 15 x$ and $15 x$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (\frac{0 - 5 + 24}{- 3 x - 1})}$

Subtract $5$ from $0$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (\frac{- 5 + 24}{- 3 x - 1})}$

Add $- 5$ and $24$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (\frac{19}{- 3 x - 1})}$

Cancel the common factor of $- 3 x - 1$ .

Cancel the common factor.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- (- 3 x - 1) \frac{19 x}{- 3 x - 1}}{(- 3 x - 1) (\frac{19}{- 3 x - 1})}$

Rewrite the expression.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{- 1 \frac{19 x}{- 3 x - 1}}{\frac{19}{- 3 x - 1}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (- 1 \frac{19 x}{- 3 x - 1} \cdot \frac{- 3 x - 1}{19})$

Rewrite $- 1 \frac{19 x}{- 3 x - 1}$ as $- \frac{19 x}{- 3 x - 1}$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (- \frac{19 x}{- 3 x - 1} \cdot \frac{- 3 x - 1}{19})$

Cancel the common factor of $19$ .

Move the leading negative in $- \frac{19 x}{- 3 x - 1}$ into the numerator.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{- 19 x}{- 3 x - 1} \cdot \frac{- 3 x - 1}{19})$

Factor $19$ out of $- 19 x$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{19 (- x)}{- 3 x - 1} \cdot \frac{- 3 x - 1}{19})$

Cancel the common factor.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{19 (- x)}{- 3 x - 1} \cdot \frac{- 3 x - 1}{19})$

Rewrite the expression.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{- x}{- 3 x - 1} \cdot (- 3 x - 1))$

Move the negative in front of the fraction.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (- \frac{x}{- 3 x - 1} \cdot (- 3 x - 1))$

Apply the distributive property.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (- \frac{x}{- 3 x - 1} \cdot (- 3 x) - \frac{x}{- 3 x - 1} \cdot - 1)$

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

Multiply $- 3$ by $- 1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (3 (\frac{x}{- 3 x - 1}) \cdot x - \frac{x}{- 3 x - 1} \cdot - 1)$

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

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{3 x}{- 3 x - 1} \cdot x - \frac{x}{- 3 x - 1} \cdot - 1)$

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

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{3 x \cdot x}{- 3 x - 1} - \frac{x}{- 3 x - 1} \cdot - 1)$

Raise $x$ to the power of $1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{3 (x \cdot x)}{- 3 x - 1} - \frac{x}{- 3 x - 1} \cdot - 1)$

Raise $x$ to the power of $1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{3 (x \cdot x)}{- 3 x - 1} - \frac{x}{- 3 x - 1} \cdot - 1)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{3 x^{1 + 1}}{- 3 x - 1} - \frac{x}{- 3 x - 1} \cdot - 1)$

Add $1$ and $1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{3 x^{2}}{- 3 x - 1} - \frac{x}{- 3 x - 1} \cdot - 1)$

Multiply $- \frac{x}{- 3 x - 1} \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{3 x^{2}}{- 3 x - 1} + 1 (\frac{x}{- 3 x - 1}))$

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

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (\frac{3 x^{2}}{- 3 x - 1} + \frac{x}{- 3 x - 1})$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{3 x^{2} + x}{- 3 x - 1}$

Factor $x$ out of $3 x^{2} + x$ .

Factor $x$ out of $3 x^{2}$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (3 x) + x}{- 3 x - 1}$

Raise $x$ to the power of $1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (3 x) + x}{- 3 x - 1}$

Factor $x$ out of $x^{1}$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (3 x) + x \cdot 1}{- 3 x - 1}$

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

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (3 x + 1)}{- 3 x - 1}$

Cancel the common factor of $3 x + 1$ and $- 3 x - 1$ .

Factor $- 1$ out of $3 x$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (- (- 3 x) + 1)}{- 3 x - 1}$

Rewrite $1$ as $- 1 (- 1)$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (- (- 3 x) - 1 \cdot - 1)}{- 3 x - 1}$

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

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (- (- 3 x - 1))}{- 3 x - 1}$

Rewrite $- (- 3 x - 1)$ as $- 1 (- 3 x - 1)$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (- 1 (- 3 x - 1))}{- 3 x - 1}$

Cancel the common factor.

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - \frac{x (- 1 (- 3 x - 1))}{- 3 x - 1}$

Divide $x \cdot (- 1)$ by $1$ .

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (x \cdot (- 1))$

Simplify the expression.

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

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = - (- 1 x)$

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

$f^{- 1} (\frac{5 x + 8}{- 3 x - 1}) = x$

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

Set up the composite result function.

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

Evaluate $f (- \frac{8 + x}{3 x + 5})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{5 (- \frac{8 + x}{3 x + 5}) + 8}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Simplify the numerator.

Multiply $5 (- \frac{8 + x}{3 x + 5})$ .

Multiply $- 1$ by $5$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{- 5 \frac{8 + x}{3 x + 5} + 8}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Combine $- 5$ and $\frac{8 + x}{3 x + 5}$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{- 5 (8 + x)}{3 x + 5} + 8}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Move the negative in front of the fraction.

$f (- \frac{8 + x}{3 x + 5}) = \frac{- \frac{(5) (8 + x)}{3 x + 5} + 8}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

To write $8$ as a fraction with a common denominator, multiply by $\frac{3 x + 5}{3 x + 5}$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{- \frac{5 (8 + x)}{3 x + 5} + \frac{8 (3 x + 5)}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Combine the numerators over the common denominator.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{- 5 (8 + x) + 8 (3 x + 5)}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Reorder terms.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{8 (3 x + 5) - 5 (8 + x)}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Rewrite $\frac{8 (3 x + 5) - 5 (8 + x)}{3 x + 5}$ in a factored form.

Apply the distributive property.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{8 (3 x) + 8 \cdot 5 - 5 (8 + x)}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Multiply $3$ by $8$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{24 x + 8 \cdot 5 - 5 (8 + x)}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Multiply $8$ by $5$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{24 x + 40 - 5 (8 + x)}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Apply the distributive property.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{24 x + 40 - 5 \cdot 8 - 5 x}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Multiply $- 5$ by $8$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{24 x + 40 - 40 - 5 x}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Subtract $5 x$ from $24 x$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x + 40 - 40}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Subtract $40$ from $40$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x + 0}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Add $19 x$ and $0$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- 3 (- \frac{8 + x}{3 x + 5}) - 1}$

Simplify the denominator.

Factor $- 1$ out of $- 3 (- \frac{8 + x}{3 x + 5}) - 1$ .

Reorder $8$ and $x$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- 3 \cdot (- 1 \frac{x + 8}{3 x + 5}) - 1}$

Factor $- 1$ out of $- 3 \cdot - 1 \frac{x + 8}{3 x + 5}$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- (- 3 \frac{x + 8}{3 x + 5}) - 1}$

Rewrite $- 1$ as $- 1 (1)$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- (- 3 \frac{x + 8}{3 x + 5}) - 1 \cdot 1}$

Factor $- 1$ out of $- (- 3 \frac{x + 8}{3 x + 5}) - 1 (1)$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- (- 3 \frac{x + 8}{3 x + 5} + 1)}$

Combine $- 3$ and $\frac{x + 8}{3 x + 5}$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- (\frac{- 3 (x + 8)}{3 x + 5} + 1)}$

Move the negative in front of the fraction.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- (- \frac{(3) (x + 8)}{3 x + 5} + 1)}$

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

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- (- \frac{3 (x + 8)}{3 x + 5} + \frac{3 x + 5}{3 x + 5})}$

Combine the numerators over the common denominator.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- \frac{- 3 (x + 8) + 3 x + 5}{3 x + 5}}$

Reorder terms.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- \frac{3 x + 5 - 3 (x + 8)}{3 x + 5}}$

Rewrite $\frac{3 x + 5 - 3 (x + 8)}{3 x + 5}$ in a factored form.

Apply the distributive property.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- \frac{3 x + 5 - 3 x - 3 \cdot 8}{3 x + 5}}$

Multiply $- 3$ by $8$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- \frac{3 x + 5 - 3 x - 24}{3 x + 5}}$

Subtract $3 x$ from $3 x$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- \frac{0 + 5 - 24}{3 x + 5}}$

Add $0$ and $5$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- \frac{5 - 24}{3 x + 5}}$

Subtract $24$ from $5$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{- \frac{- 19}{3 x + 5}}$

Move the negative in front of the fraction.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{1 (\frac{19}{3 x + 5})}$

Combine exponents.

Factor out negative.

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{\frac{19}{3 x + 5}}$

Multiply $- 1$ by $- 1$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{1 (\frac{19}{3 x + 5})}$

Multiply $\frac{19}{3 x + 5}$ by $1$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{\frac{19 x}{3 x + 5}}{\frac{19}{3 x + 5}}$

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{8 + x}{3 x + 5}) = \frac{19 x}{3 x + 5} \cdot \frac{3 x + 5}{19}$

Cancel the common factor of $19$ .

Factor $19$ out of $19 x$ .

$f (- \frac{8 + x}{3 x + 5}) = \frac{19 (x)}{3 x + 5} \cdot \frac{3 x + 5}{19}$

Cancel the common factor.

$f (- \frac{8 + x}{3 x + 5}) = \frac{19 x}{3 x + 5} \cdot \frac{3 x + 5}{19}$

Rewrite the expression.

$f (- \frac{8 + x}{3 x + 5}) = \frac{x}{3 x + 5} \cdot (3 x + 5)$

Cancel the common factor of $3 x + 5$ .

Cancel the common factor.

$f (- \frac{8 + x}{3 x + 5}) = \frac{x}{3 x + 5} \cdot (3 x + 5)$

Rewrite the expression.

$f (- \frac{8 + x}{3 x + 5}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{8 + x}{3 x + 5}$ is the inverse of $f (x) = \frac{5 x + 8}{- 3 x - 1}$ .

$f^{- 1} (x) = - \frac{8 + x}{3 x + 5}$

Find the Inverse (5x+8)/(-3x-1)

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