To find the inverse of the function [tex]\( f(x) = 2x + 1 \)[/tex], follow these steps:
1. Express the function in terms of [tex]\( y \)[/tex]:
[tex]\[
y = 2x + 1
\][/tex]
2. Swap [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[
x = 2y + 1
\][/tex]
3. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
x - 1 = 2y
\][/tex]
[tex]\[
y = \frac{x - 1}{2}
\][/tex]
4. Rewrite the inverse function:
[tex]\[
f^{-1}(x) = \frac{x - 1}{2}
\][/tex]
5. Compare the form of the inverse function with the options provided:
- [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]
6. The correct form of the inverse function [tex]\(\frac{x - 1}{2}\)[/tex] can be rewritten as:
[tex]\[
\frac{x - 1}{2} = \frac{1}{2} x - \frac{1}{2}
\][/tex]
Therefore, the inverse function is:
[tex]\[ h(x) = \frac{1}{2} x - \frac{1}{2} \][/tex]
From the given options, this corresponds to the first function.
So, the correct choice is:
[tex]\[ \boxed{h(x) = \frac{1}{2}x - \frac{1}{2}} \][/tex]
