To find the inverse function [tex]\( f^{-1}(x) \)[/tex] for the given function [tex]\( f(x) = 2x + 5 \)[/tex], follow these steps:
1. Start with the function [tex]\( f(x) = 2x + 5 \)[/tex]. Replace [tex]\( f(x) \)[/tex] by [tex]\( y \)[/tex]
[tex]\[
y = 2x + 5
\][/tex]
2. Swap the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
[tex]\[
x = 2y + 5
\][/tex]
3. Solve for [tex]\( y \)[/tex]
- Subtract 5 from both sides:
[tex]\[
x - 5 = 2y
\][/tex]
- Divide both sides by 2 to isolate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{x - 5}{2}
\][/tex]
So, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{x - 5}{2}
\][/tex]
4. Find [tex]\( f^{-1}(8) \)[/tex]
Substitute [tex]\( x = 8 \)[/tex] into the inverse function:
[tex]\[
f^{-1}(8) = \frac{8 - 5}{2} = \frac{3}{2}
\][/tex]
Therefore, the value of [tex]\( f^{-1}(8) \)[/tex] is [tex]\( \frac{3}{2} \)[/tex].
So the correct answer is [tex]\( \frac{3}{2} \)[/tex].
