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Which function is the inverse of [tex]\( f(x) = 2x + 3 \)[/tex]?

A. [tex]\( f^{-1}(x) = -\frac{1}{2}x - \frac{3}{2} \)[/tex]
B. [tex]\( f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} \)[/tex]
C. [tex]\( f^{-1}(x) = -2x + 3 \)[/tex]
D. [tex]\( f^{-1}(x) = 2x + 3 \)[/tex]

Sagot :

To find the inverse of the function [tex]\( f(x) = 2x + 3 \)[/tex], we need to follow a systematic process. An inverse function essentially "undoes" what the original function does.

Here are the steps to find the inverse function:

1. Express the function as an equation in terms of [tex]\( y \)[/tex]:
[tex]\[ y = 2x + 3 \][/tex]

2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to create the equation of the inverse:
[tex]\[ x = 2y + 3 \][/tex]

3. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ x = 2y + 3 \][/tex]

Subtract 3 from both sides:
[tex]\[ x - 3 = 2y \][/tex]

Divide both sides by 2:
[tex]\[ y = \frac{x - 3}{2} \][/tex]

Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{x - 3}{2} \][/tex]

Let's compare with the given options and choose the correct inverse function:

1. [tex]\( f^{-1}(x) = -\frac{1}{2}x - \frac{3}{2} \)[/tex]
2. [tex]\( f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} \)[/tex]
3. [tex]\( f^{-1}(x) = -2x + 3 \)[/tex]
4. [tex]\( f^{-1}(x) = 2x + 3 \)[/tex]

We can see that the correct inverse function [tex]\( \frac{x - 3}{2} \)[/tex] can be rewritten as:

[tex]\[ f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} \][/tex]

Therefore, the correct answer is:
[tex]\[ f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} \][/tex]

So the correct option is:

[tex]\[ \boxed{f^{-1}(x) = \frac{1}{2}x - \frac{3}{2}} \][/tex]