Colon Ortiz
05/17/2024 · Primary School
For \( f(x)=3 x-3 \) and \( g(x)=4 x^{2}-3 \), find the following functions. a. \( (f \circ g)(x) \); b. \( (g \circ f)(x) ; \) c. \( (f \circ g)(-2) ; \) d. \( (g \circ f)(-2) \)
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Tutor-Verified Answer
Step-by-step Solution
To find the compositions of the functions \( f(x) = 3x - 3 \) and \( g(x) = 4x^2 - 3 \), we will follow the definitions of function composition.
### a. \( (f \circ g)(x) \)
The composition \( (f \circ g)(x) \) means we will substitute \( g(x) \) into \( f(x) \):
\[
g(x) = 4x^2 - 3
\]
Now substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f(4x^2 - 3) = 3(4x^2 - 3) - 3
\]
Now simplify:
\[
= 12x^2 - 9 - 3 = 12x^2 - 12
\]
Thus,
\[
(f \circ g)(x) = 12x^2 - 12
\]
### b. \( (g \circ f)(x) \)
The composition \( (g \circ f)(x) \) means we will substitute \( f(x) \) into \( g(x) \):
\[
f(x) = 3x - 3
\]
Now substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(3x - 3) = 4(3x - 3)^2 - 3
\]
Now calculate \( (3x - 3)^2 \):
\[
(3x - 3)^2 = 9x^2 - 18x + 9
\]
Now substitute this back into \( g \):
\[
g(f(x)) = 4(9x^2 - 18x + 9) - 3 = 36x^2 - 72x + 36 - 3
\]
Now simplify:
\[
= 36x^2 - 72x + 33
\]
Thus,
\[
(g \circ f)(x) = 36x^2 - 72x + 33
\]
### c. \( (f \circ g)(-2) \)
Now we will evaluate \( (f \circ g)(-2) \):
First, find \( g(-2) \):
\[
g(-2) = 4(-2)^2 - 3 = 4(4) - 3 = 16 - 3 = 13
\]
Now substitute this into \( f \):
\[
f(g(-2)) = f(13) = 3(13) - 3 = 39 - 3 = 36
\]
Thus,
\[
(f \circ g)(-2) = 36
\]
### d. \( (g \circ f)(-2) \)
Now we will evaluate \( (g \circ f)(-2) \):
First, find \( f(-2) \):
\[
f(-2) = 3(-2) - 3 = -6 - 3 = -9
\]
Now substitute this into \( g \):
\[
g(f(-2)) = g(-9) = 4(-9)^2 - 3 = 4(81) - 3 = 324 - 3 = 321
\]
Thus,
\[
(g \circ f)(-2) = 321
\]
### Summary of Results
- \( (f \circ g)(x) = 12x^2 - 12 \)
- \( (g \circ f)(x) = 36x^2 - 72x + 33 \)
- \( (f \circ g)(-2) = 36 \)
- \( (g \circ f)(-2) = 321 \)
Quick Answer
- (f ∘ g)(x) = 12x^2 - 12
- (g ∘ f)(x) = 36x^2 - 72x + 33
- (f ∘ g)(-2) = 36
- (g ∘ f)(-2) = 321
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