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]
