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What is the inverse of the function [tex][tex]$f(x)=2x+1$[/tex][/tex]?

A. [tex][tex]$h(x)=\frac{1}{2}x-\frac{1}{2}$[/tex][/tex]
B. [tex][tex]$h(x)=\frac{1}{2}x+\frac{1}{2}$[/tex][/tex]
C. [tex][tex]$h(x)=\frac{1}{2}x-2$[/tex][/tex]
D. [tex][tex]$h(x)=\frac{1}{2}x+2$[/tex][/tex]


Sagot :

To find the inverse of the function [tex]\(f(x) = 2x + 1\)[/tex], follow these steps:

1. Start with the given function:
[tex]\[ f(x) = 2x + 1 \][/tex]

2. Replace [tex]\(f(x)\)[/tex] with [tex]\(y\)[/tex] for easier manipulation:
[tex]\[ y = 2x + 1 \][/tex]

3. Solve for [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]. First, isolate the term with [tex]\(x\)[/tex]:
[tex]\[ y - 1 = 2x \][/tex]

4. Next, solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{y - 1}{2} \][/tex]

5. Now replace [tex]\(y\)[/tex] with [tex]\(x\)[/tex] to write the inverse function:
[tex]\[ h(x) = \frac{x - 1}{2} \][/tex]

6. Simplify the expression:
[tex]\[ h(x) = \frac{1}{2}x - \frac{1}{2} \][/tex]

This matches the form given in one of the options. Therefore, the inverse function is:
[tex]\[ h(x) = \frac{1}{2}x - \frac{1}{2} \][/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 answer is:
[tex]\[ h(x) = \frac{1}{2}x - \frac{1}{2} \][/tex]