Find the Inverse f(x)=x-1

$f (x) = x - 1$

Write $f (x) = x - 1$ as an equation.

$y = x - 1$

Interchange the variables.

$x = y - 1$

Solve for $y$ .

Rewrite the equation as $y - 1 = x$ .

$y - 1 = x$

Add $1$ to both sides of the equation.

$y = x + 1$

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

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

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

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

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

Combine the opposite terms in $(x - 1) + 1$ .

Add $- 1$ and $1$ .

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

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

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

$f (x + 1) = (x + 1) - 1$

Combine the opposite terms in $(x + 1) - 1$ .

Subtract $1$ from $1$ .

$f (x + 1) = x + 0$

Add $x$ and $0$ .

$f (x + 1) = x$

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

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