Fuentes Robbins
01/18/2023 · Primary School
Given that \( f(x)=-2 x-6 \) and \( g(x)=2 x^{2}+8 x-2 \), calculate (a) \( f(g(-2))=\square \) (b) \( g(f(4))=\square \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To solve the problem, we will first calculate \( g(-2) \) and then use that result to find \( f(g(-2)) \). After that, we will calculate \( f(4) \) and use that result to find \( g(f(4)) \).
### Part (a): Calculate \( f(g(-2)) \)
1. **Calculate \( g(-2) \)**:
\[
g(x) = 2x^2 + 8x - 2
\]
Substitute \( x = -2 \):
\[
g(-2) = 2(-2)^2 + 8(-2) - 2
\]
\[
= 2(4) - 16 - 2
\]
\[
= 8 - 16 - 2
\]
\[
= -10
\]
2. **Calculate \( f(g(-2)) = f(-10) \)**:
\[
f(x) = -2x - 6
\]
Substitute \( x = -10 \):
\[
f(-10) = -2(-10) - 6
\]
\[
= 20 - 6
\]
\[
= 14
\]
Thus, \( f(g(-2)) = 14 \).
### Part (b): Calculate \( g(f(4)) \)
1. **Calculate \( f(4) \)**:
\[
f(x) = -2x - 6
\]
Substitute \( x = 4 \):
\[
f(4) = -2(4) - 6
\]
\[
= -8 - 6
\]
\[
= -14
\]
2. **Calculate \( g(f(4)) = g(-14) \)**:
\[
g(x) = 2x^2 + 8x - 2
\]
Substitute \( x = -14 \):
\[
g(-14) = 2(-14)^2 + 8(-14) - 2
\]
\[
= 2(196) - 112 - 2
\]
\[
= 392 - 112 - 2
\]
\[
= 392 - 114
\]
\[
= 278
\]
Thus, \( g(f(4)) = 278 \).
### Final Answers:
(a) \( f(g(-2)) = 14 \)
(b) \( g(f(4)) = 278 \)
Quick Answer
(a) \( f(g(-2)) = 14 \)
(b) \( g(f(4)) = 278 \)
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