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

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


Sagot :

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]