Find the Inverse g(x)=2x+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 $2 y + 4 = x$ .

$2 y + 4 = x$

Subtract $4$ from both sides of the equation.

$2 y = x - 4$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Divide $- 4$ by $2$ .

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

Step 4

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

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

Step 5

Verify if $g^{- 1} (x) = \frac{x}{2} - 2$ 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 $g^{- 1}$ .

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

Cancel the common factor of $2 x + 4$ and $2$ .

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $4$ .

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

Factor $2$ out of $2 (x) + 2 (2)$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $x + 2$ by $1$ .

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

Combine the opposite terms in $x + 2 - 2$ .

Subtract $2$ from $2$ .

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

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

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

Simplify each term.

Apply the distributive property.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $2$ by $- 2$ .

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

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

Add $- 4$ and $4$ .

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

Add $x$ and $0$ .

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

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

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

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

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