Find the Inverse 1/3x+4

$\frac{1}{3} x + 4$

Step 1

Interchange the variables.

$x = \frac{1}{3} y + 4$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{1}{3} y + 4 = x$ .

$\frac{1}{3} y + 4 = x$

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

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

Subtract $4$ from both sides of the equation.

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

Multiply both sides of the equation by $3$ .

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

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$y = 3 (x - 4)$

Simplify the right side.

Simplify $3 (x - 4)$ .

Apply the distributive property.

$y = 3 x + 3 \cdot - 4$

Multiply $3$ by $- 4$ .

$y = 3 x - 12$

Step 3

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

$f^{- 1} (x) = 3 x - 12$

Step 4

Verify if $f^{- 1} (x) = 3 x - 12$ is the inverse of $f (x) = \frac{1}{3} x + 4$ .

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

$f^{- 1} (\frac{1}{3} x + 4) = 3 (\frac{1}{3} x + 4) - 12$

Simplify each term.

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

$f^{- 1} (\frac{1}{3} x + 4) = 3 (\frac{x}{3} + 4) - 12$

Apply the distributive property.

$f^{- 1} (\frac{1}{3} x + 4) = 3 (\frac{x}{3}) + 3 \cdot 4 - 12$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (\frac{1}{3} x + 4) = 3 (\frac{x}{3}) + 3 \cdot 4 - 12$

Rewrite the expression.

$f^{- 1} (\frac{1}{3} x + 4) = x + 3 \cdot 4 - 12$

Multiply $3$ by $4$ .

$f^{- 1} (\frac{1}{3} x + 4) = x + 12 - 12$

Combine the opposite terms in $x + 12 - 12$ .

Subtract $12$ from $12$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

Evaluate $f (3 x - 12)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (3 x - 12) = \frac{1}{3} \cdot (3 x - 12) + 4$

Simplify each term.

Apply the distributive property.

$f (3 x - 12) = \frac{1}{3} \cdot (3 x) + \frac{1}{3} \cdot - 12 + 4$

Cancel the common factor of $3$ .

Factor $3$ out of $3 x$ .

$f (3 x - 12) = \frac{1}{3} \cdot (3 (x)) + \frac{1}{3} \cdot - 12 + 4$

Cancel the common factor.

$f (3 x - 12) = \frac{1}{3} \cdot (3 x) + \frac{1}{3} \cdot - 12 + 4$

Rewrite the expression.

$f (3 x - 12) = x + \frac{1}{3} \cdot - 12 + 4$

Cancel the common factor of $3$ .

Factor $3$ out of $- 12$ .

$f (3 x - 12) = x + \frac{1}{3} \cdot (3 (- 4)) + 4$

Cancel the common factor.

$f (3 x - 12) = x + \frac{1}{3} \cdot (3 \cdot - 4) + 4$

Rewrite the expression.

$f (3 x - 12) = x - 4 + 4$

Combine the opposite terms in $x - 4 + 4$ .

Add $- 4$ and $4$ .

$f (3 x - 12) = x + 0$

Add $x$ and $0$ .

$f (3 x - 12) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 3 x - 12$ is the inverse of $f (x) = \frac{1}{3} x + 4$ .

$f^{- 1} (x) = 3 x - 12$

Find the Inverse 1/3x+4

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