Sure, let's solve the given problems step-by-step.
1. Understanding the given values:
- We are given that [tex]\( f^{-1}(-2) = 0.5 \)[/tex]. This means when inputting [tex]\(-2\)[/tex] into the inverse function [tex]\( f^{-1} \)[/tex], the output is [tex]\(0.5\)[/tex].
- We are also given that [tex]\( f(-4) = -4 \)[/tex]. This means when inputting [tex]\(-4\)[/tex] into the function [tex]\( f \)[/tex], the output is [tex]\(-4\)[/tex].
- We are asked to find [tex]\( f(f^{-1}(-2)) \)[/tex].
2. Finding [tex]\( f(f^{-1}(-2)) \)[/tex]:
- By the definition of inverse functions, [tex]\( f \)[/tex] and [tex]\( f^{-1} \)[/tex] cancel each other out. Specifically:
[tex]\[
f(f^{-1}(x)) = x
\][/tex]
for any [tex]\(x\)[/tex] in the domain of [tex]\(f^{-1}\)[/tex].
3. Applying this property:
- We need to evaluate [tex]\( f(f^{-1}(-2)) \)[/tex].
- By the property of inverse functions, substituting [tex]\(-2\)[/tex] into the expression, we get:
[tex]\[
f(f^{-1}(-2)) = -2
\][/tex]
So, let's summarize the results:
- [tex]\( f^{-1}(-2) = 0.5 \)[/tex]
- [tex]\( f(-4) = -4 \)[/tex]
- [tex]\( f(f^{-1}(-2)) = -2 \)[/tex]
Thus, the final answer to the problems is:
[tex]\[
(0.5, -4, -2)
\][/tex]
