Sure, let's solve these step-by-step.
1. Calculate [tex]\( f(2) \)[/tex]:
Using the function [tex]\( f(x) = \frac{1}{2} x \)[/tex],
[tex]\[
f(2) = \frac{1}{2} \times 2 = 1
\][/tex]
So,
[tex]\[
f(2) = 1.0
\][/tex]
2. Calculate [tex]\( f^{-1}(1) \)[/tex]:
Using the inverse function [tex]\( f^{-1}(x) = 2 x \)[/tex],
[tex]\[
f^{-1}(1) = 2 \times 1 = 2
\][/tex]
So,
[tex]\[
f^{-1}(1) = 2
\][/tex]
3. Calculate [tex]\( f^{-1}(f(2)) \)[/tex]:
First, let's find [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 1
\][/tex]
Then, apply the inverse function to the result:
[tex]\[
f^{-1}(f(2)) = f^{-1}(1) = 2
\][/tex]
So,
[tex]\[
f^{-1}(f(2)) = 2.0
\][/tex]
4. Calculate [tex]\( f^{-1}(-2) \)[/tex]:
Using the inverse function [tex]\( f^{-1}(x) = 2 x \)[/tex],
[tex]\[
f^{-1}(-2) = 2 \times -2 = -4
\][/tex]
So,
[tex]\[
f^{-1}(-2) = -4
\][/tex]
5. Calculate [tex]\( f(-4) \)[/tex]:
Using the function [tex]\( f(x) = \frac{1}{2} x \)[/tex],
[tex]\[
f(-4) = \frac{1}{2} \times -4 = -2
\][/tex]
So,
[tex]\[
f(-4) = -2.0
\][/tex]
6. Calculate [tex]\( f(f^{-1}(-2)) \)[/tex]:
First, let's find [tex]\( f^{-1}(-2) \)[/tex]:
[tex]\[
f^{-1}(-2) = -4
\][/tex]
Then, apply the function to the result:
[tex]\[
f(f^{-1}(-2)) = f(-4) = -2
\][/tex]
So,
[tex]\[
f(f^{-1}(-2)) = -2.0
\][/tex]
Combining all the calculated results, we have:
[tex]\[
\begin{array}{l}
f(2) = 1.0 \\
f^{-1}(1) = 2 \\
f^{-1}(f(2)) = 2.0 \\
f^{-1}(-2) = -4 \\
f(-4) = -2.0 \\
f(f^{-1}(-2)) = -2.0
\end{array}
\][/tex]
