Find the Inverse g(x)=x+3

$g (x) = x + 3$

Step 1

Write $g (x) = x + 3$ as an equation.

$y = x + 3$

Step 2

Interchange the variables.

$x = y + 3$

Step 3

Solve for $y$ .

Rewrite the equation as $y + 3 = x$ .

$y + 3 = x$

Subtract $3$ from both sides of the equation.

$y = x - 3$

Step 4

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

$g^{- 1} (x) = x - 3$

Step 5

Verify if $g^{- 1} (x) = x - 3$ is the inverse of $g (x) = x + 3$ .

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

$g^{- 1} (x + 3) = (x + 3) - 3$

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

Subtract $3$ from $3$ .

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

Add $x$ and $0$ .

$g^{- 1} (x + 3) = x$

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

Set up the composite result function.

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

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

$g (x - 3) = (x - 3) + 3$

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

Add $- 3$ and $3$ .

$g (x - 3) = x + 0$

Add $x$ and $0$ .

$g (x - 3) = x$

Since $g^{- 1} (g (x)) = x$ and $g (g^{- 1} (x)) = x$ , then $g^{- 1} (x) = x - 3$ is the inverse of $g (x) = x + 3$ .

$g^{- 1} (x) = x - 3$