Find the Inverse f(x)=3x-12

$f (x) = 3 x - 12$

Step 1

Write $f (x) = 3 x - 12$ as an equation.

$y = 3 x - 12$

Step 2

Interchange the variables.

$x = 3 y - 12$

Step 3

Solve for $y$ .

Rewrite the equation as $3 y - 12 = x$ .

$3 y - 12 = x$

Add $12$ to both sides of the equation.

$3 y = x + 12$

Divide each term in $3 y = x + 12$ by $3$ and simplify.

Divide each term in $3 y = x + 12$ by $3$ .

$\frac{3 y}{3} = \frac{x}{3} + \frac{12}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} = \frac{x}{3} + \frac{12}{3}$

Divide $y$ by $1$ .

$y = \frac{x}{3} + \frac{12}{3}$

Simplify the right side.

Divide $12$ by $3$ .

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

Step 4

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

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

Step 5

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

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

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

Cancel the common factor of $3 x - 12$ and $3$ .

Factor $3$ out of $3 x$ .

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

Factor $3$ out of $- 12$ .

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

Factor $3$ out of $3 (x) + 3 (- 4)$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $x - 4$ by $1$ .

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

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

Add $- 4$ and $4$ .

$f^{- 1} (3 x - 12) = x + 0$

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $4$ .

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

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

Subtract $12$ from $12$ .

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

Add $x$ and $0$ .

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

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

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

Find the Inverse f(x)=3x-12

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