To find the inverse of the function [tex]\( f(x) = 2x + 1 \)[/tex], we need to follow several specific steps:
1. Rewrite the Function's Equation:
Let's begin with the function:
[tex]\[
y = f(x) = 2x + 1
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
To find the inverse function, we swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. This leads to:
[tex]\[
x = 2y + 1
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
Now, solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
x = 2y + 1
\][/tex]
Subtract 1 from both sides to isolate the term with [tex]\( y \)[/tex]:
[tex]\[
x - 1 = 2y
\][/tex]
Divide both sides by 2 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{x - 1}{2}
\][/tex]
4. Express the Inverse Function:
The expression [tex]\( y = \frac{x - 1}{2} \)[/tex] is the inverse of the original function. Thus, we can write:
[tex]\[
h(x) = \frac{x - 1}{2}
\][/tex]
5. Simplify the Inverse Function:
Simplifying [tex]\( h(x) = \frac{x - 1}{2} \)[/tex] gives:
[tex]\[
h(x) = \frac{1}{2} x - \frac{1}{2}
\][/tex]
Given the four choices:
1. [tex]\( h(x) = \frac{1}{2} x - \frac{1}{2} \)[/tex]
2. [tex]\( h(x) = \frac{1}{2} x + \frac{1}{2} \)[/tex]
3. [tex]\( h(x) = \frac{1}{2} x - 2 \)[/tex]
4. [tex]\( h(x) = \frac{1}{2} x + 2 \)[/tex]
The correct inverse function is:
[tex]\[
h(x) = \frac{1}{2} x - \frac{1}{2}
\][/tex]
Therefore, the inverse of the function [tex]\( f(x) = 2x + 1 \)[/tex] is:
[tex]\[
h(x) = \frac{1}{2} x - \frac{1}{2}
\][/tex]
Which corresponds to the first choice. Hence, the answer is:
[tex]\[
\boxed{1}
\][/tex]
