To find the inverse of the function [tex]\( f(x) = 2x + 3 \)[/tex], we need to follow these steps:
1. Rewrite the function in terms of [tex]\( y \)[/tex]:
[tex]\[ y = 2x + 3 \][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y = 2x + 3 \][/tex]
Subtract 3 from both sides:
[tex]\[ y - 3 = 2x \][/tex]
Divide both sides by 2:
[tex]\[ x = \frac{y - 3}{2} \][/tex]
3. Express the inverse function [tex]\( f^{-1}(x) \)[/tex]:
To rewrite our equation properly in the inverse function format, replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex]:
[tex]\[ f^{-1}(x) = \frac{x - 3}{2} \][/tex]
4. Simplify the inverse function:
[tex]\[ f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} \][/tex]
Now, let's compare this function with the given options:
- [tex]\( f^{-1}(x) = -\frac{1}{2}x - \frac{3}{2} \)[/tex]
- [tex]\( f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} \)[/tex]
- [tex]\( f^{-1}(x) = -2x + 3 \)[/tex]
- [tex]\( f^{-1}(x) = 2x + 3 \)[/tex]
The correctly simplified inverse function is:
[tex]\[ f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} \][/tex]
Hence, the correct answer is:
[tex]\[ f^{-1}(x) = \frac{1}{2}x - \frac{3}{2} \][/tex]
