Find the Inverse g(x)=x-4

$g (x) = x - 4$

Step 1

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

$y = x - 4$

Step 2

Interchange the variables.

$x = y - 4$

Step 3

Solve for $y$ .

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

$y - 4 = x$

Add $4$ to both sides of the equation.

$y = x + 4$

Step 4

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

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

Step 5

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

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

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

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

Add $- 4$ and $4$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

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

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

Subtract $4$ from $4$ .

$g (x + 4) = x + 0$

Add $x$ and $0$ .

$g (x + 4) = x$

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

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