Question
For f left parenthesis x right parenthesis equals 3 minus x and g left parenthesis x right parenthesis equals 3 x squared plus x plus 5 comma find the following functions.
a. left parenthesis f circle g right parenthesis left parenthesis x right parenthesis; b. left parenthesis g circle f right parenthesis left parenthesis x right parenthesis; c. left parenthesis f circle g right parenthesis left parenthesis 2 right parenthesis; d. left parenthesis g circle f right parenthesis left parenthesis 2 right parenthesis
Question content area bottom
Part 1
a. left parenthesis f circle g right parenthesis left parenthesis x right parenthesisequals
enter your response here
(Simplify your answer.)
a. left parenthesis f circle g right parenthesis left parenthesis x right parenthesis; b. left parenthesis g circle f right parenthesis left parenthesis x right parenthesis; c. left parenthesis f circle g right parenthesis left parenthesis 2 right parenthesis; d. left parenthesis g circle f right parenthesis left parenthesis 2 right parenthesis
Question content area bottom
Part 1
a. left parenthesis f circle g right parenthesis left parenthesis x right parenthesisequals
enter your response here
(Simplify your answer.)
Answers
GPT-4o mini
To find the composite functions \( (f \circ g)(x) \) and \( (g \circ f)(x) \), we need to substitute one function into the other. Let's go through each part step-by-step.
### Part a: \( (f \circ g)(x) \)
This means we need to calculate \( f(g(x)) \).
1. First, we want to find \( g(x) \):
\[
g(x) = 3x^2 + x + 5
\]
2. Now, substitute \( g(x) \) into \( f(x) \):
\[
f(x) = 3 - x
\]
Thus,
\[
f(g(x)) = f(3x^2 + x + 5) = 3 - (3x^2 + x + 5)
\]
3. Simplifying \( f(g(x)) \):
\[
= 3 - 3x^2 - x - 5
\]
\[
= -3x^2 - x - 2
\]
So,
\[
(f \circ g)(x) = -3x^2 - x - 2
\]
### Part b: \( (g \circ f)(x) \)
This means we need to calculate \( g(f(x)) \).
1. First, find \( f(x) \):
\[
f(x) = 3 - x
\]
2. Now substitute \( f(x) \) into \( g(x) \):
\[
g(x) = 3x^2 + x + 5
\]
Thus,
\[
g(f(x)) = g(3 - x) = 3(3 - x)^2 + (3 - x) + 5
\]
3. Expanding \( (3 - x)^2 \):
\[
(3 - x)^2 = 9 - 6x + x^2
\]
Therefore,
\[
g(f(x)) = 3(9 - 6x + x^2) + (3 - x) + 5
\]
\[
= 27 - 18x + 3x^2 + 3 - x + 5
\]
\[
= 3x^2 - 19x + 35
\]
So,
\[
(g \circ f)(x) = 3x^2 - 19x + 35
\]
### Part c: \( (f \circ g)(2) \)
We can substitute \( x = 2 \) into \( (f \circ g)(x) \):
\[
(f \circ g)(2) = -3(2^2) - 2 - 2
\]
\[
= -3(4) - 2 - 2
\]
\[
= -12 - 2 - 2
\]
\[
= -16
\]
### Part d: \( (g \circ f)(2) \)
Now, we substitute \( x = 2 \) into \( (g \circ f)(x) \):
\[
(g \circ f)(2) = 3(2^2) - 19(2) + 35
\]
\[
= 3(4) - 38 + 35
\]
\[
= 12 - 38 + 35
\]
\[
= 12 - 3
\]
\[
= 9
\]
### Final Answers:
a. \( (f \circ g)(x) = -3x^2 - x - 2 \)
b. \( (g \circ f)(x) = 3x^2 - 19x + 35 \)
c. \( (f \circ g)(2) = -16 \)
d. \( (g \circ f)(2) = 9 \)
### Part a: \( (f \circ g)(x) \)
This means we need to calculate \( f(g(x)) \).
1. First, we want to find \( g(x) \):
\[
g(x) = 3x^2 + x + 5
\]
2. Now, substitute \( g(x) \) into \( f(x) \):
\[
f(x) = 3 - x
\]
Thus,
\[
f(g(x)) = f(3x^2 + x + 5) = 3 - (3x^2 + x + 5)
\]
3. Simplifying \( f(g(x)) \):
\[
= 3 - 3x^2 - x - 5
\]
\[
= -3x^2 - x - 2
\]
So,
\[
(f \circ g)(x) = -3x^2 - x - 2
\]
### Part b: \( (g \circ f)(x) \)
This means we need to calculate \( g(f(x)) \).
1. First, find \( f(x) \):
\[
f(x) = 3 - x
\]
2. Now substitute \( f(x) \) into \( g(x) \):
\[
g(x) = 3x^2 + x + 5
\]
Thus,
\[
g(f(x)) = g(3 - x) = 3(3 - x)^2 + (3 - x) + 5
\]
3. Expanding \( (3 - x)^2 \):
\[
(3 - x)^2 = 9 - 6x + x^2
\]
Therefore,
\[
g(f(x)) = 3(9 - 6x + x^2) + (3 - x) + 5
\]
\[
= 27 - 18x + 3x^2 + 3 - x + 5
\]
\[
= 3x^2 - 19x + 35
\]
So,
\[
(g \circ f)(x) = 3x^2 - 19x + 35
\]
### Part c: \( (f \circ g)(2) \)
We can substitute \( x = 2 \) into \( (f \circ g)(x) \):
\[
(f \circ g)(2) = -3(2^2) - 2 - 2
\]
\[
= -3(4) - 2 - 2
\]
\[
= -12 - 2 - 2
\]
\[
= -16
\]
### Part d: \( (g \circ f)(2) \)
Now, we substitute \( x = 2 \) into \( (g \circ f)(x) \):
\[
(g \circ f)(2) = 3(2^2) - 19(2) + 35
\]
\[
= 3(4) - 38 + 35
\]
\[
= 12 - 38 + 35
\]
\[
= 12 - 3
\]
\[
= 9
\]
### Final Answers:
a. \( (f \circ g)(x) = -3x^2 - x - 2 \)
b. \( (g \circ f)(x) = 3x^2 - 19x + 35 \)
c. \( (f \circ g)(2) = -16 \)
d. \( (g \circ f)(2) = 9 \)