To find the inverse of the function [tex]\( f(x) = 2x - 4 \)[/tex], we need to follow these steps:
1. Express the function as [tex]\( y = 2x - 4 \)[/tex]:
[tex]\[
y = 2x - 4
\][/tex]
2. Swap the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = 2y - 4
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[
x = 2y - 4
\][/tex]
Add 4 to both sides:
[tex]\[
x + 4 = 2y
\][/tex]
Divide by 2:
[tex]\[
y = \frac{x + 4}{2}
\][/tex]
Simplify:
[tex]\[
y = \frac{1}{2}x + 2
\][/tex]
So, the inverse function is:
[tex]\[
f^{-1}(x) = \frac{1}{2}x + 2
\][/tex]
Now let's match this with the provided options:
1. [tex]\( g(x) = \frac{1}{2} x - \frac{1}{4} \)[/tex]
2. [tex]\( g(x) = \frac{1}{4} x - \frac{1}{2} \)[/tex]
3. [tex]\( g(x) = 4x + 2 \)[/tex]
4. [tex]\( g(x) = \frac{1}{2} x + 2 \)[/tex]
From these options, the correct inverse function that matches [tex]\(\frac{1}{2}x + 2\)[/tex] is:
[tex]\[
g(x) = \frac{1}{2} x + 2
\][/tex]
Therefore, the correct answer is the fourth option:
[tex]\[
g(x) = \frac{1}{2} x + 2
\][/tex]
