To determine the inverse of the function [tex]\( f(x) = 2x + 1 \)[/tex], we follow these steps:
1. Express the function as an equation in [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[
y = 2x + 1
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to solve for the inverse function:
[tex]\[
x = 2y + 1
\][/tex]
3. Solve this equation for [tex]\( y \)[/tex]:
To isolate [tex]\( y \)[/tex]:
[tex]\[
x - 1 = 2y
\][/tex]
Dividing both sides by 2:
[tex]\[
y = \frac{x - 1}{2}
\][/tex]
4. Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = \frac{1}{2} x - \frac{1}{2}
\][/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{1}{2} x - \frac{1}{2}
\][/tex]
Given the choices:
- [tex]\( h(x) = \frac{1}{2} x - \frac{1}{2} \)[/tex]
- [tex]\( h(x) = \frac{1}{2} x + \frac{1}{2} \)[/tex]
- [tex]\( h(x) = \frac{1}{2} x - 2 \)[/tex]
- [tex]\( h(x) = \frac{1}{2} x + 2 \)[/tex]
The correct choice is:
[tex]\[
h(x) = \frac{1}{2} x - \frac{1}{2}
\][/tex]
