Find the Inverse g(x)=-2(x-4)

$g (x) = - 2 (x - 4)$

Step 1

Write $g (x) = - 2 (x - 4)$ as an equation.

$y = - 2 (x - 4)$

Step 2

Interchange the variables.

$x = - 2 (y - 4)$

Step 3

Solve for $y$ .

Rewrite the equation as $- 2 (y - 4) = x$ .

$- 2 (y - 4) = x$

Divide each term in $- 2 (y - 4) = x$ by $- 2$ and simplify.

Divide each term in $- 2 (y - 4) = x$ by $- 2$ .

$\frac{- 2 (y - 4)}{- 2} = \frac{x}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 (y - 4)}{- 2} = \frac{x}{- 2}$

Divide $y - 4$ by $1$ .

$y - 4 = \frac{x}{- 2}$

Simplify the right side.

Move the negative in front of the fraction.

$y - 4 = - \frac{x}{2}$

Add $4$ to both sides of the equation.

$y = - \frac{x}{2} + 4$

Step 4

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

$g^{- 1} (x) = - \frac{x}{2} + 4$

Step 5

Verify if $g^{- 1} (x) = - \frac{x}{2} + 4$ is the inverse of $g (x) = - 2 (x - 4)$ .

To verify the inverse, check if $g^{- 1} (g (x)) = x$ and $g (g^{- 1} (x)) = x$ .

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

Set up the composite result function.

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

Evaluate $g^{- 1} (- 2 (x - 4))$ by substituting in the value of $g$ into $g^{- 1}$ .

$g^{- 1} (- 2 (x - 4)) = - \frac{- 2 (x - 4)}{2} + 4$

Simplify each term.

Cancel the common factor of $- 2$ and $2$ .

Factor $2$ out of $- 2 (x - 4)$ .

$g^{- 1} (- 2 (x - 4)) = - \frac{2 (- (x - 4))}{2} + 4$

Cancel the common factors.

Factor $2$ out of $2$ .

$g^{- 1} (- 2 (x - 4)) = - \frac{2 (- (x - 4))}{2 (1)} + 4$

Cancel the common factor.

$g^{- 1} (- 2 (x - 4)) = - \frac{2 (- (x - 4))}{2 \cdot 1} + 4$

Rewrite the expression.

$g^{- 1} (- 2 (x - 4)) = - \frac{- (x - 4)}{1} + 4$

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

$g^{- 1} (- 2 (x - 4)) = (x - 4) + 4$

Apply the distributive property.

$g^{- 1} (- 2 (x - 4)) = - (- x + 4) + 4$

Multiply $- 1$ by $- 4$ .

$g^{- 1} (- 2 (x - 4)) = - (- x + 4) + 4$

Apply the distributive property.

$g^{- 1} (- 2 (x - 4)) = x - 1 \cdot 4 + 4$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$g^{- 1} (- 2 (x - 4)) = 1 x - 1 \cdot 4 + 4$

Multiply $x$ by $1$ .

$g^{- 1} (- 2 (x - 4)) = x - 1 \cdot 4 + 4$

Multiply $- 1$ by $4$ .

$g^{- 1} (- 2 (x - 4)) = x - 4 + 4$

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

Add $- 4$ and $4$ .

$g^{- 1} (- 2 (x - 4)) = x + 0$

Add $x$ and $0$ .

$g^{- 1} (- 2 (x - 4)) = x$

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

Set up the composite result function.

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

Evaluate $g (- \frac{x}{2} + 4)$ by substituting in the value of $g^{- 1}$ into $g$ .

$g (- \frac{x}{2} + 4) = - 2 ((- \frac{x}{2} + 4) - 4)$

Combine the opposite terms in $(- \frac{x}{2} + 4) - 4$ .

Subtract $4$ from $4$ .

$g (- \frac{x}{2} + 4) = - 2 (- \frac{x}{2} + 0)$

Add $- \frac{x}{2}$ and $0$ .

$g (- \frac{x}{2} + 4) = - 2 (- \frac{x}{2})$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{x}{2}$ into the numerator.

$g (- \frac{x}{2} + 4) = - 2 \frac{- x}{2}$

Factor $2$ out of $- 2$ .

$g (- \frac{x}{2} + 4) = 2 (- 1) (\frac{- x}{2})$

Cancel the common factor.

$g (- \frac{x}{2} + 4) = 2 \cdot (- 1 \frac{- x}{2})$

Rewrite the expression.

$g (- \frac{x}{2} + 4) = - 1 (- x)$

Multiply.

Multiply $- 1$ by $- 1$ .

$g (- \frac{x}{2} + 4) = 1 x$

Multiply $x$ by $1$ .

$g (- \frac{x}{2} + 4) = x$

Since $g^{- 1} (g (x)) = x$ and $g (g^{- 1} (x)) = x$ , then $g^{- 1} (x) = - \frac{x}{2} + 4$ is the inverse of $g (x) = - 2 (x - 4)$ .

$g^{- 1} (x) = - \frac{x}{2} + 4$

Find the Inverse g(x)=-2(x-4)

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