To answer the given question, let's break it down into two parts: finding the inverse function [tex]\( f^{-1}(x) \)[/tex] and evaluating it at a specific point.
### Part (a): Finding [tex]\( f^{-1}(x) \)[/tex]
Given the function [tex]\( f(x) = \frac{x^3 - 7}{2} \)[/tex], we need to find its inverse function [tex]\( f^{-1}(x) \)[/tex].
1. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
Start by writing the function in terms of [tex]\( y \)[/tex]:
[tex]\[ y = \frac{x^3 - 7}{2} \][/tex]
2. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
Interchange the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \frac{y^3 - 7}{2} \][/tex]
3. Solve for [tex]\( y \)[/tex]:
Now, solve the above equation for [tex]\( y \)[/tex]:
[tex]\[ 2x = y^3 - 7 \][/tex]
[tex]\[ y^3 = 2x + 7 \][/tex]
[tex]\[ y = \sqrt[3]{2x + 7} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = (2x + 7)^{1/3} \][/tex]
### Part (b): Finding [tex]\( f^{-1}(28.5) \)[/tex]
Now, we need to evaluate the inverse function at [tex]\( x = 28.5 \)[/tex].
Substitute [tex]\( x = 28.5 \)[/tex] into the inverse function:
[tex]\[ f^{-1}(28.5) = (2 \cdot 28.5 + 7)^{1/3} \][/tex]
[tex]\[ f^{-1}(28.5) = (57 + 7)^{1/3} \][/tex]
[tex]\[ f^{-1}(28.5) = 64^{1/3} \][/tex]
[tex]\[ f^{-1}(28.5) = 4 \][/tex]
Therefore, the solution to the given parts are:
a) [tex]\[ f^{-1}(x) = (2x + 7)^{1/3} \][/tex]
b) [tex]\[ f^{-1}(28.5) = 4 \][/tex]
