Certainly! Let's work through the problem step-by-step.
### Part (a): Find [tex]\( f(4) \)[/tex]
Given the function [tex]\( f(x) = \frac{x^3}{4} + 6 \)[/tex], we need to find [tex]\( f(4) \)[/tex].
1. Substitute [tex]\( x = 4 \)[/tex] into the function:
[tex]\[
f(4) = \frac{4^3}{4} + 6
\][/tex]
2. Calculate [tex]\( 4^3 \)[/tex]:
[tex]\[
4^3 = 64
\][/tex]
3. Substitute [tex]\( 64 \)[/tex] back into the function:
[tex]\[
f(4) = \frac{64}{4} + 6
\][/tex]
4. Simplify [tex]\(\frac{64}{4}\)[/tex]:
[tex]\[
\frac{64}{4} = 16
\][/tex]
5. Add [tex]\( 16 \)[/tex] and [tex]\( 6 \)[/tex]:
[tex]\[
16 + 6 = 22
\][/tex]
So, [tex]\( f(4) = 22 \)[/tex].
### Part (b): Find [tex]\( f^{-1}(x) \)[/tex]
To find the inverse function [tex]\( f^{-1}(x) \)[/tex], we need to solve the equation [tex]\( y = f(x) \)[/tex] for [tex]\( x \)[/tex].
1. Start with the equation:
[tex]\[
y = \frac{x^3}{4} + 6
\][/tex]
2. Subtract 6 from both sides to isolate the term involving [tex]\( x \)[/tex]:
[tex]\[
y - 6 = \frac{x^3}{4}
\][/tex]
3. Multiply both sides by 4 to get rid of the denominator:
[tex]\[
4(y - 6) = x^3
\][/tex]
4. Solve for [tex]\( x \)[/tex] by taking the cube root of both sides:
[tex]\[
x = \sqrt[3]{4(y - 6)}
\][/tex]
So, the inverse function is:
[tex]\[
f^{-1}(x) = (4x - 24)^{1/3}
\][/tex]
### Part (c): Find [tex]\( f^{-1}(8) \)[/tex]
Now we need to evaluate [tex]\( f^{-1}(8) \)[/tex].
1. Substitute [tex]\( x = 8 \)[/tex] into the inverse function [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
f^{-1}(8) = (4(8) - 24)^{1/3}
\][/tex]
2. Calculate [tex]\( 4 \times 8 \)[/tex]:
[tex]\[
4 \times 8 = 32
\][/tex]
3. Subtract 24 from 32:
[tex]\[
32 - 24 = 8
\][/tex]
4. Take the cube root of 8:
[tex]\[
8^{1/3} = 2
\][/tex]
Therefore, [tex]\( f^{-1}(8) = 2 \)[/tex].
Thus, the results are:
- [tex]\( f(4) = 22 \)[/tex]
- [tex]\( f^{-1}(x) = (4x - 24)^{1/3} \)[/tex]
- [tex]\( f^{-1}(8) = 2 \)[/tex]
