Question

find the inverse of the function $f(x) = 2x - 4$.
$g(x) = \frac{1}{2}x - \frac{1}{4}$
$g(x) = \frac{1}{4}x - \frac{1}{2}$
$g(x) = 4x + 2$
$g(x) = \frac{1}{2}x + 2$

Explanation:

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

We start with the function \( f(x) = 2x - 4 \). Replace \( f(x) \) with \( y \), so we have \( y = 2x - 4 \).

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

To find the inverse, we swap the roles of \( x \) and \( y \). This gives us \( x = 2y - 4 \).

Step3: Solve for \( y \)

First, add 4 to both sides of the equation: \( x + 4 = 2y \). Then, divide both sides by 2: \( y=\frac{x + 4}{2}=\frac{1}{2}x+2 \). So the inverse function \( g(x)=\frac{1}{2}x + 2 \).

Answer:

\( g(x)=\frac{1}{2}x + 2 \) (the last option: \( g(x)=\frac{1}{2}x+2 \))