To find the inverse of the function [tex]\( f(x) = 2x + 3 \)[/tex], follow these steps:
1. Write the function in terms of [tex]\( y \)[/tex]:
[tex]\[
y = 2x + 3
\][/tex]
2. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[
x = 2y + 3
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[
x - 3 = 2y
\][/tex]
[tex]\[
y = \frac{x - 3}{2}
\][/tex]
4. Rewrite [tex]\( y \)[/tex] as [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
f^{-1}(x) = \frac{x - 3}{2}
\][/tex]
Thus, the inverse function is [tex]\( f^{-1}(x) = \frac{x - 3}{2} \)[/tex].
Now, let's compare this with the given options:
- Option 1: [tex]\( f^{-1}(x) = -\frac{1}{2}x - \frac{3}{2} \)[/tex]
- Option 2: [tex]\( f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} \)[/tex]
- Option 3: [tex]\( f^{-1}(x) = -2x + 3 \)[/tex]
- Option 4: [tex]\( f^{-1}(x) = 2x + 3 \)[/tex]
Clearly, the correct option that matches our derived inverse function is:
[tex]\[ f^{-1}(x) = \frac{1}{2}(x - 3) \][/tex]
Simplifying further,
[tex]\[ f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} \][/tex]
So, the correct answer is:
[tex]\[ f^{-1}(x) = \frac{1}{2} x - \frac{3}{2} \][/tex]
Therefore, the correct option is:
[tex]\[ f^{-1}(x) = \frac{1}{2} x - \frac{3}{2} \][/tex]
