To find the inverse of the function [tex]\( f(x) = 2x + 1 \)[/tex], let's go through the process step-by-step:
1. Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
Start by writing the function in terms of [tex]\( y \)[/tex],
[tex]\[
y = f(x) = 2x + 1.
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
To find the inverse function, swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex],
[tex]\[
x = 2y + 1.
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
Now solve the equation for [tex]\( y \)[/tex].
- Subtract 1 from both sides,
[tex]\[
x - 1 = 2y.
\][/tex]
- Divide both sides by 2,
[tex]\[
y = \frac{x - 1}{2}.
\][/tex]
4. Simplify the expression:
Simplify the right-hand side of the equation,
[tex]\[
y = \frac{x}{2} - \frac{1}{2}.
\][/tex]
So, the inverse function [tex]\( h(x) \)[/tex] is:
[tex]\[
h(x) = \frac{x}{2} - \frac{1}{2}.
\][/tex]
Therefore, the correct choice is:
[tex]\[
h(x) = \frac{1}{2} x - \frac{1}{2}.
\][/tex]
