Find the Inverse -1/3x+8

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

Rewrite the equation as $- \frac{1}{3} y + 8 = x$ .

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

Combine $y$ and $\frac{1}{3}$ .

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

Subtract $8$ from both sides of the equation.

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

Multiply both sides of the equation by $- 3$ .

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

Simplify both sides of the equation.

Simplify the left side.

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

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{y}{3}$ into the numerator.

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

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply.

Multiply $- 1$ by $- 1$ .

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

Multiply $y$ by $1$ .

$y = - 3 (x - 8)$

Simplify the right side.

Simplify $- 3 (x - 8)$ .

Apply the distributive property.

$y = - 3 x - 3 \cdot - 8$

Multiply $- 3$ by $- 8$ .

$y = - 3 x + 24$

Step 3

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

$f^{- 1} (x) = - 3 x + 24$

Step 4

Verify if $f^{- 1} (x) = - 3 x + 24$ is the inverse of $f (x) = - \frac{1}{3} x + 8$ .

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

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

Simplify each term.

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

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

Apply the distributive property.

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

Cancel the common factor of $3$ .

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

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

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Multiply $- 3$ by $8$ .

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

Combine the opposite terms in $x - 24 + 24$ .

Add $- 24$ and $24$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

Evaluate $f (- 3 x + 24)$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $3$ .

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Cancel the common factor of $3$ .

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

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

Factor $3$ out of $24$ .

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

Cancel the common factor.

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

Rewrite the expression.

$f (- 3 x + 24) = x - 1 \cdot 8 + 8$

Multiply $- 1$ by $8$ .

$f (- 3 x + 24) = x - 8 + 8$

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

Add $- 8$ and $8$ .

$f (- 3 x + 24) = x + 0$

Add $x$ and $0$ .

$f (- 3 x + 24) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - 3 x + 24$ is the inverse of $f (x) = - \frac{1}{3} x + 8$ .

$f^{- 1} (x) = - 3 x + 24$

Find the Inverse -1/3x+8

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