Find the Inverse f(x)=x-6

$f (x) = x - 6$

Step 1

Write $f (x) = x - 6$ as an equation.

$y = x - 6$

Step 2

Interchange the variables.

$x = y - 6$

Step 3

Solve for $y$ .

Rewrite the equation as $y - 6 = x$ .

$y - 6 = x$

Add $6$ to both sides of the equation.

$y = x + 6$

Step 4

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

$f^{- 1} (x) = x + 6$

Step 5

Verify if $f^{- 1} (x) = x + 6$ is the inverse of $f (x) = x - 6$ .

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 $f^{- 1} (x - 6)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (x - 6) = (x - 6) + 6$

Combine the opposite terms in $(x - 6) + 6$ .

Add $- 6$ and $6$ .

$f^{- 1} (x - 6) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (x - 6) = x$

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

Set up the composite result function.

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

Evaluate $f (x + 6)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x + 6) = (x + 6) - 6$

Combine the opposite terms in $(x + 6) - 6$ .

Subtract $6$ from $6$ .

$f (x + 6) = x + 0$

Add $x$ and $0$ .

$f (x + 6) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x + 6$ is the inverse of $f (x) = x - 6$ .

$f^{- 1} (x) = x + 6$