Let's solve the given problems step by step using the functions [tex]\( f(x) = \frac{1}{2} x \)[/tex] and [tex]\( f^{-1}(x) = 2x \)[/tex].
1. Calculate [tex]\( f(2) \)[/tex]:
[tex]\[
f(x) = \frac{1}{2} x
\][/tex]
Substitute [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = \frac{1}{2} \times 2 = 1.0
\][/tex]
So,
[tex]\[
f(2) = 1.0
\][/tex]
2. Calculate [tex]\( f^{-1}(1) \)[/tex]:
[tex]\[
f^{-1}(x) = 2x
\][/tex]
Substitute [tex]\( x = 1 \)[/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]:
We already know from the previous calculation that:
[tex]\[
f(2) = 1.0
\][/tex]
Now, substitute [tex]\( x = 1.0 \)[/tex] into [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
f^{-1}(1.0) = 2 \times 1.0 = 2.0
\][/tex]
So,
[tex]\[
f^{-1}(f(2)) = 2.0
\][/tex]
### Summary of the results:
[tex]\[
f(2) = 1.0
\][/tex]
[tex]\[
f^{-1}(1) = 2
\][/tex]
[tex]\[
f^{-1}(f(2)) = 2.0
\][/tex]
