Question

examples 2 and 3
find the inverse of each function.

1. $f(x) = x + 2$

$x = y + 2$
$\frac{x - 2}{2} = \frac{y}{2}$

Explanation:

Step1: Replace \( f(x) \) with \( y \)

We start by writing the function as \( y = x + 2 \). This is a standard first step when finding the inverse of a function, as it helps us work with the equation in terms of \( x \) and \( y \) more easily.

Step2: Swap \( x \) and \( y \)

Next, we interchange the roles of \( x \) and \( y \) to get \( x = y + 2 \). This is based on the definition of an inverse function, where if \( y = f(x) \), then \( x = f^{-1}(y) \), so swapping \( x \) and \( y \) allows us to solve for the inverse function.

Step3: Solve for \( y \)

To solve for \( y \), we subtract 2 from both sides of the equation \( x = y + 2 \). This gives us \( y = x - 2 \).

Step4: Replace \( y \) with \( f^{-1}(x) \)

Finally, we replace \( y \) with the notation for the inverse function, \( f^{-1}(x) \). So the inverse function is \( f^{-1}(x) = x - 2 \).

Answer:

The inverse of the function \( f(x) = x + 2 \) is \( f^{-1}(x) = x - 2 \)