To find the inverse of the function [tex]\( f(x) = 2x + 3 \)[/tex], we can follow a systematic step-by-step approach. The goal is to find a function [tex]\( f^{-1}(x) \)[/tex] such that when you apply [tex]\( f \)[/tex] to [tex]\( f^{-1}(x) \)[/tex], you get [tex]\( x \)[/tex].
Here are the steps to find the inverse function:
1. Rewrite [tex]\( f(x) \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y = 2x + 3 \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to reflect finding the inverse:
[tex]\[ x = 2y + 3 \][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[ x - 3 = 2y \][/tex]
Divide both sides by 2:
[tex]\[ y = \frac{x - 3}{2} \][/tex]
Simplify the expression:
[tex]\[ y = \frac{1}{2}x - \frac{3}{2} \][/tex]
Now that we have [tex]\( y \)[/tex] expressed in terms of [tex]\( x \)[/tex], we can write the inverse function as:
[tex]\[ f^{-1}(x) = \frac{1}{2} x - \frac{3}{2} \][/tex]
Among the given choices, the correct inverse function is:
[tex]\[ f^{-1}(x) = \frac{1}{2} x - \frac{3}{2} \][/tex]
Thus, the corresponding option number is:
[tex]\[ 2 \][/tex]
So the correct answer is:
[tex]\[ f^{-1}(x)=\frac{1}{2} x-\frac{3}{2} \][/tex]
