Question

find the inverse of the function $f(x) = 2x - 4$.
$g(x)=\frac{1}{2}x+\frac{1}{4}$
$g(x)=\frac{1}{4}x+\frac{1}{2}$
$g(x)=4x + 2$
$g(x)=\frac{1}{2}x + 2$
because $f(x)$ dropdown, its inverse is a function.
done
complete

Explanation:

Step1: Find the inverse of \( f(x) = 2x - 4 \)

To find the inverse, we first replace \( f(x) \) with \( y \), so \( y = 2x - 4 \). Then we swap \( x \) and \( y \): \( x = 2y - 4 \). Now we solve for \( y \):

1. Add 4 to both sides: \( x + 4 = 2y \)
2. Divide both sides by 2: \( y = \frac{1}{2}x + 2 \)

So the inverse function \( g(x) = \frac{1}{2}x + 2 \), which matches the checked option.

Step2: Determine why the inverse is a function

A function has an inverse that is also a function if it is one - to - one (passes the horizontal line test). The function \( f(x)=2x - 4 \) is a linear function with a slope \( m = 2
eq0 \). All linear functions with non - zero slopes are one - to - one (since a horizontal line will intersect a non - vertical line at most once). So \( f(x) \) is one - to - one, and thus its inverse is a function.

Answer:

The correct inverse function is \( g(x)=\frac{1}{2}x + 2 \) (the option with the checkmark). And \( f(x) \) is one - to - one (a linear function with non - zero slope), so its inverse is a function.