Find the Inverse f^-1(x)=x+10

$f^{- 1} (x) = x + 10$

Step 1

Write $f^{- 1} (x) = x + 10$ as an equation.

$y = x + 10$

Step 2

Interchange the variables.

$x = y + 10$

Step 3

Solve for $y$ .

Rewrite the equation as $y + 10 = x$ .

$y + 10 = x$

Subtract $10$ from both sides of the equation.

$y = x - 10$

Step 4

Replace $y$ with $f (x)$ to show the final answer.

$f (x) = x - 10$

Step 5

Verify if $f (x) = x - 10$ is the inverse of $f^{- 1} (x) = x + 10$ .

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

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

Set up the composite result function.

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

Evaluate $f (x + 10)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x + 10) = (x + 10) - 10$

Combine the opposite terms in $(x + 10) - 10$ .

Subtract $10$ from $10$ .

$f (x + 10) = x + 0$

Add $x$ and $0$ .

$f (x + 10) = x$

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

Set up the composite result function.

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

Evaluate $f^{- 1} (x - 10)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (x - 10) = (x - 10) + 10$

Combine the opposite terms in $(x - 10) + 10$ .

Add $- 10$ and $10$ .

$f^{- 1} (x - 10) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (x - 10) = x$

Since $f (f^{- 1} (x)) = x$ and $f^{- 1} (f (x)) = x$ , then $f (x) = x - 10$ is the inverse of $f^{- 1} (x) = x + 10$ .

$f (x) = x - 10$