To find the inverse of the function [tex]\( f(x) = 2x + 1 \)[/tex], we need to follow these steps:
1. Start by writing the function:
[tex]\[
y = 2x + 1
\][/tex]
2. Swap [tex]\( y \)[/tex] and [tex]\( x \)[/tex] to find the inverse function:
[tex]\[
x = 2y + 1
\][/tex]
3. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
- Subtract 1 from both sides:
[tex]\[
x - 1 = 2y
\][/tex]
- Divide both sides by 2:
[tex]\[
y = \frac{x - 1}{2}
\][/tex]
4. Rewrite [tex]\( y \)[/tex] as [tex]\( h(x) \)[/tex], which represents the inverse function:
[tex]\[
h(x) = \frac{x - 1}{2}
\][/tex]
5. Simplify the expression to match it with the options provided:
[tex]\[
h(x) = \frac{1}{2}x - \frac{1}{2}
\][/tex]
The inverse of the function [tex]\( f(x) = 2x + 1 \)[/tex] is:
[tex]\[
h(x) = \frac{1}{2}x - \frac{1}{2}
\][/tex]
Thus, the correct answer is:
[tex]\[
h(x) = \frac{1}{2}x - \frac{1}{2}
\][/tex]
