Find the Inverse (3x)/4+8

$\frac{3 x}{4} + 8$

Step 1

Interchange the variables.

$x = \frac{3 y}{4} + 8$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{3 y}{4} + 8 = x$ .

$\frac{3 y}{4} + 8 = x$

Subtract $8$ from both sides of the equation.

$\frac{3 y}{4} = x - 8$

Multiply both sides of the equation by $\frac{4}{3}$ .

$\frac{4}{3} \cdot \frac{3 y}{4} = \frac{4}{3} (x - 8)$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{4}{3} \cdot \frac{3 y}{4}$ .

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 y}{4} = \frac{4}{3} (x - 8)$

Rewrite the expression.

$\frac{1}{3} (3 y) = \frac{4}{3} (x - 8)$

Cancel the common factor of $3$ .

Factor $3$ out of $3 y$ .

$\frac{1}{3} (3 (y)) = \frac{4}{3} (x - 8)$

Cancel the common factor.

$\frac{1}{3} (3 y) = \frac{4}{3} (x - 8)$

Rewrite the expression.

$y = \frac{4}{3} (x - 8)$

Simplify the right side.

Simplify $\frac{4}{3} (x - 8)$ .

Apply the distributive property.

$y = \frac{4}{3} x + \frac{4}{3} \cdot - 8$

Combine $\frac{4}{3}$ and $x$ .

$y = \frac{4 x}{3} + \frac{4}{3} \cdot - 8$

Multiply $\frac{4}{3} \cdot - 8$ .

Combine $\frac{4}{3}$ and $- 8$ .

$y = \frac{4 x}{3} + \frac{4 \cdot - 8}{3}$

Multiply $4$ by $- 8$ .

$y = \frac{4 x}{3} + \frac{- 32}{3}$

Move the negative in front of the fraction.

$y = \frac{4 x}{3} - \frac{32}{3}$

Step 3

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

$f^{- 1} (x) = \frac{4 x}{3} - \frac{32}{3}$

Step 4

Verify if $f^{- 1} (x) = \frac{4 x}{3} - \frac{32}{3}$ is the inverse of $f (x) = \frac{3 x}{4} + 8$ .

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 $f^{- 1} (\frac{3 x}{4} + 8)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{3 x}{4} + 8) = \frac{4 (\frac{3 x}{4} + 8)}{3} - \frac{32}{3}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{3 x}{4} + 8) = \frac{4 (\frac{3 x}{4} + 8) - 32}{3}$

Simplify each term.

Apply the distributive property.

$f^{- 1} (\frac{3 x}{4} + 8) = \frac{4 (\frac{3 x}{4}) + 4 \cdot 8 - 32}{3}$

Cancel the common factor of $4$ .

Cancel the common factor.

$f^{- 1} (\frac{3 x}{4} + 8) = \frac{4 (\frac{3 x}{4}) + 4 \cdot 8 - 32}{3}$

Rewrite the expression.

$f^{- 1} (\frac{3 x}{4} + 8) = \frac{3 x + 4 \cdot 8 - 32}{3}$

Multiply $4$ by $8$ .

$f^{- 1} (\frac{3 x}{4} + 8) = \frac{3 x + 32 - 32}{3}$

Simplify terms.

Combine the opposite terms in $3 x + 32 - 32$ .

Subtract $32$ from $32$ .

$f^{- 1} (\frac{3 x}{4} + 8) = \frac{3 x + 0}{3}$

Add $3 x$ and $0$ .

$f^{- 1} (\frac{3 x}{4} + 8) = \frac{3 x}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (\frac{3 x}{4} + 8) = \frac{3 x}{3}$

Divide $x$ by $1$ .

$f^{- 1} (\frac{3 x}{4} + 8) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{4 x}{3} - \frac{32}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{4 x}{3} - \frac{32}{3}) = \frac{3 (\frac{4 x}{3} - \frac{32}{3})}{4} + 8$

Simplify each term.

Simplify the numerator.

Combine the numerators over the common denominator.

$f (\frac{4 x}{3} - \frac{32}{3}) = \frac{3 (\frac{4 x - 32}{3})}{4} + 8$

Factor $4$ out of $4 x - 32$ .

Factor $4$ out of $4 x$ .

$f (\frac{4 x}{3} - \frac{32}{3}) = \frac{3 (\frac{4 (x) - 32}{3})}{4} + 8$

Factor $4$ out of $- 32$ .

$f (\frac{4 x}{3} - \frac{32}{3}) = \frac{3 (\frac{4 x + 4 \cdot - 8}{3})}{4} + 8$

Factor $4$ out of $4 x + 4 \cdot - 8$ .

$f (\frac{4 x}{3} - \frac{32}{3}) = \frac{3 (\frac{4 (x - 8)}{3})}{4} + 8$

Combine $3$ and $\frac{4 (x - 8)}{3}$ .

$f (\frac{4 x}{3} - \frac{32}{3}) = \frac{\frac{3 (4 (x - 8))}{3}}{4} + 8$

Multiply $3$ by $4$ .

$f (\frac{4 x}{3} - \frac{32}{3}) = \frac{\frac{12 (x - 8)}{3}}{4} + 8$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{12 (x - 8)}{3}$ by cancelling the common factors.

Factor $3$ out of $12 (x - 8)$ .

$f (\frac{4 x}{3} - \frac{32}{3}) = \frac{\frac{3 (4 (x - 8))}{3}}{4} + 8$

Factor $3$ out of $3$ .

$f (\frac{4 x}{3} - \frac{32}{3}) = \frac{\frac{3 (4 (x - 8))}{3 (1)}}{4} + 8$

Cancel the common factor.

$f (\frac{4 x}{3} - \frac{32}{3}) = \frac{\frac{3 (4 (x - 8))}{3 \cdot 1}}{4} + 8$

Rewrite the expression.

$f (\frac{4 x}{3} - \frac{32}{3}) = \frac{\frac{4 (x - 8)}{1}}{4} + 8$

Divide $4 (x - 8)$ by $1$ .

$f (\frac{4 x}{3} - \frac{32}{3}) = \frac{4 (x - 8)}{4} + 8$

Cancel the common factor of $4$ .

Cancel the common factor.

$f (\frac{4 x}{3} - \frac{32}{3}) = \frac{4 (x - 8)}{4} + 8$

Divide $x - 8$ by $1$ .

$f (\frac{4 x}{3} - \frac{32}{3}) = x - 8 + 8$

Combine the opposite terms in $x - 8 + 8$ .

Add $- 8$ and $8$ .

$f (\frac{4 x}{3} - \frac{32}{3}) = x + 0$

Add $x$ and $0$ .

$f (\frac{4 x}{3} - \frac{32}{3}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{4 x}{3} - \frac{32}{3}$ is the inverse of $f (x) = \frac{3 x}{4} + 8$ .

$f^{- 1} (x) = \frac{4 x}{3} - \frac{32}{3}$

Find the Inverse (3x)/4+8

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