Find the Inverse f(x)=-4x-12

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

Step 1

Write $f (x) = - 4 x - 12$ as an equation.

$y = - 4 x - 12$

Step 2

Interchange the variables.

$x = - 4 y - 12$

Step 3

Solve for $y$ .

Rewrite the equation as $- 4 y - 12 = x$ .

$- 4 y - 12 = x$

Add $12$ to both sides of the equation.

$- 4 y = x + 12$

Divide each term in $- 4 y = x + 12$ by $- 4$ and simplify.

Divide each term in $- 4 y = x + 12$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{x}{- 4} + \frac{12}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{x}{- 4} + \frac{12}{- 4}$

Divide $y$ by $1$ .

$y = \frac{x}{- 4} + \frac{12}{- 4}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{x}{4} + \frac{12}{- 4}$

Divide $12$ by $- 4$ .

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

Step 4

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

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

Step 5

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

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 $f^{- 1} (- 4 x - 12)$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Simplify each term.

Cancel the common factor of $- 4 x - 12$ and $4$ .

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

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

Factor $4$ out of $- 12$ .

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

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

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

Cancel the common factors.

Factor $4$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- x - 3$ by $1$ .

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

Apply the distributive property.

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

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Multiply $- 1$ by $- 3$ .

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

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

Subtract $3$ from $3$ .

$f^{- 1} (- 4 x - 12) = x + 0$

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{x}{4}$ into the numerator.

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

Factor $4$ out of $- 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Multiply $- 4$ by $- 3$ .

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

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

Subtract $12$ from $12$ .

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

Add $x$ and $0$ .

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

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

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

Find the Inverse f(x)=-4x-12

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