Answer :
To solve the problem, we are given two functions: [tex]\( f(x) = x^2 - 2 \)[/tex] and [tex]\( h(x) = x^3 - 1 \)[/tex]. We need to find the value of [tex]\( f(h(-2)) \)[/tex].
Here's the step-by-step process:
1. Calculate [tex]\( h(-2) \)[/tex]:
- Start with the function [tex]\( h(x) = x^3 - 1 \)[/tex].
- Substitute [tex]\( x = -2 \)[/tex] into the function:
[tex]\[
h(-2) = (-2)^3 - 1
\][/tex]
- Calculate [tex]\( (-2)^3 = -8 \)[/tex], therefore:
[tex]\[
h(-2) = -8 - 1 = -9
\][/tex]
2. Calculate [tex]\( f(h(-2)) \)[/tex], which is [tex]\( f(-9) \)[/tex]:
- Use the function [tex]\( f(x) = x^2 - 2 \)[/tex].
- Substitute [tex]\( x = -9 \)[/tex] into the function:
[tex]\[
f(-9) = (-9)^2 - 2
\][/tex]
- Calculate [tex]\( (-9)^2 = 81 \)[/tex], therefore:
[tex]\[
f(-9) = 81 - 2 = 79
\][/tex]
So, the final answer is [tex]\( f(h(-2)) = 79 \)[/tex].
Here's the step-by-step process:
1. Calculate [tex]\( h(-2) \)[/tex]:
- Start with the function [tex]\( h(x) = x^3 - 1 \)[/tex].
- Substitute [tex]\( x = -2 \)[/tex] into the function:
[tex]\[
h(-2) = (-2)^3 - 1
\][/tex]
- Calculate [tex]\( (-2)^3 = -8 \)[/tex], therefore:
[tex]\[
h(-2) = -8 - 1 = -9
\][/tex]
2. Calculate [tex]\( f(h(-2)) \)[/tex], which is [tex]\( f(-9) \)[/tex]:
- Use the function [tex]\( f(x) = x^2 - 2 \)[/tex].
- Substitute [tex]\( x = -9 \)[/tex] into the function:
[tex]\[
f(-9) = (-9)^2 - 2
\][/tex]
- Calculate [tex]\( (-9)^2 = 81 \)[/tex], therefore:
[tex]\[
f(-9) = 81 - 2 = 79
\][/tex]
So, the final answer is [tex]\( f(h(-2)) = 79 \)[/tex].
