Luna Cervantes
09/05/2023 · High School
Given that \( f(x)=-3 x^{2}-5 x \) and \( g(x)=-4 x-9 \), determine each of the following. Make sure to fully simplify your answer. (a) \( (f \circ g)(x)=\square \) (b) \( (g \circ f)(x)=\square \) (c) \( (f \circ f)(x)=\square \) (d) \( (g \circ g)(x)= \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To solve the problems, we will compute the compositions of the functions \( f(x) \) and \( g(x) \).
Given:
\[
f(x) = -3x^2 - 5x
\]
\[
g(x) = -4x - 9
\]
### (a) \( (f \circ g)(x) \)
This means we need to compute \( f(g(x)) \):
\[
g(x) = -4x - 9
\]
Now substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(-4x - 9) = -3(-4x - 9)^2 - 5(-4x - 9)
\]
First, calculate \( (-4x - 9)^2 \):
\[
(-4x - 9)^2 = 16x^2 + 72x + 81
\]
Now substitute this back into \( f \):
\[
f(-4x - 9) = -3(16x^2 + 72x + 81) - 5(-4x - 9)
\]
\[
= -48x^2 - 216x - 243 + 20x + 45
\]
Combine like terms:
\[
= -48x^2 - 196x - 198
\]
Thus,
\[
(f \circ g)(x) = -48x^2 - 196x - 198
\]
### (b) \( (g \circ f)(x) \)
This means we need to compute \( g(f(x)) \):
\[
f(x) = -3x^2 - 5x
\]
Now substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(-3x^2 - 5x) = -4(-3x^2 - 5x) - 9
\]
\[
= 12x^2 + 20x - 9
\]
Thus,
\[
(g \circ f)(x) = 12x^2 + 20x - 9
\]
### (c) \( (f \circ f)(x) \)
This means we need to compute \( f(f(x)) \):
\[
f(x) = -3x^2 - 5x
\]
Now substitute \( f(x) \) into itself:
\[
f(f(x)) = f(-3x^2 - 5x) = -3(-3x^2 - 5x)^2 - 5(-3x^2 - 5x)
\]
First, calculate \( (-3x^2 - 5x)^2 \):
\[
(-3x^2 - 5x)^2 = 9x^4 + 30x^3 + 25x^2
\]
Now substitute this back into \( f \):
\[
f(-3x^2 - 5x) = -3(9x^4 + 30x^3 + 25x^2) + 15x^2 + 25x
\]
\[
= -27x^4 - 90x^3 - 75x^2 + 15x^2 + 25x
\]
Combine like terms:
\[
= -27x^4 - 90x^3 - 60x^2 + 25x
\]
Thus,
\[
(f \circ f)(x) = -27x^4 - 90x^3 - 60x^2 + 25x
\]
### (d) \( (g \circ g)(x) \)
This means we need to compute \( g(g(x)) \):
\[
g(x) = -4x - 9
\]
Now substitute \( g(x) \) into itself:
\[
g(g(x)) = g(-4x - 9) = -4(-4x - 9) - 9
\]
\[
= 16x + 36 - 9
\]
\[
= 16x + 27
\]
Thus,
\[
(g \circ g)(x) = 16x + 27
\]
### Final Answers
(a) \( (f \circ g)(x) = -48x^2 - 196x - 198 \)
(b) \( (g \circ f)(x) = 12x^2 + 20x - 9 \)
(c) \( (f \circ f)(x) = -27x^4 - 90x^3 - 60x^2 + 25x \)
(d) \( (g \circ g)(x) = 16x + 27 \)
Quick Answer
(a) \( (f \circ g)(x) = -48x^2 - 196x - 198 \)
(b) \( (g \circ f)(x) = 12x^2 + 20x - 9 \)
(c) \( (f \circ f)(x) = -27x^4 - 90x^3 - 60x^2 + 25x \)
(d) \( (g \circ g)(x) = 16x + 27 \)
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