Question
For f left parenthesis x right parenthesis equals x squared plus 7 and g left parenthesis x right parenthesis equals x squared minus 4 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 3 right parenthesis; d. left parenthesis g circle f right parenthesis left parenthesis 3 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 3 right parenthesis; d. left parenthesis g circle f right parenthesis left parenthesis 3 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 compositions of the functions \( f \) and \( g \) defined as:
- \( f(x) = x^2 + 7 \)
- \( g(x) = x^2 - 4 \)
we need to calculate \( (f \circ g)(x) \) and \( (g \circ f)(x) \).
### Part a: \( (f \circ g)(x) \)
The composition \( (f \circ g)(x) \) means we substitute \( g(x) \) into \( f(x) \).
1. Compute \( g(x) = x^2 - 4 \).
2. Substitute \( g(x) \) into \( f \):
\[
f(g(x)) = f(x^2 - 4) = (x^2 - 4)^2 + 7
\]
Now, expand \( (x^2 - 4)^2 \):
\[
(x^2 - 4)^2 = x^4 - 8x^2 + 16
\]
Now substitute back into \( f(g(x)) \):
\[
f(g(x)) = x^4 - 8x^2 + 16 + 7
\]
Simplify:
\[
f(g(x)) = x^4 - 8x^2 + 23
\]
**Final Answer for Part a:**
\[
(f \circ g)(x) = x^4 - 8x^2 + 23
\]
---
### Part b: \( (g \circ f)(x) \)
Now compute \( (g \circ f)(x) \).
1. Compute \( f(x) = x^2 + 7 \).
2. Substitute \( f(x) \) into \( g \):
\[
g(f(x)) = g(x^2 + 7) = (x^2 + 7)^2 - 4
\]
Now, expand \( (x^2 + 7)^2 \):
\[
(x^2 + 7)^2 = x^4 + 14x^2 + 49
\]
Now substitute back into \( g(f(x)) \):
\[
g(f(x)) = x^4 + 14x^2 + 49 - 4
\]
Simplify:
\[
g(f(x)) = x^4 + 14x^2 + 45
\]
**Final Answer for Part b:**
\[
(g \circ f)(x) = x^4 + 14x^2 + 45
\]
---
### Part c: \( (f \circ g)(3) \)
Now compute \( (f \circ g)(3) \).
1. Compute \( g(3) \):
\[
g(3) = 3^2 - 4 = 9 - 4 = 5
\]
2. Now compute \( f(5) \):
\[
f(5) = 5^2 + 7 = 25 + 7 = 32
\]
**Final Answer for Part c:**
\[
(f \circ g)(3) = 32
\]
---
### Part d: \( (g \circ f)(3) \)
Now compute \( (g \circ f)(3) \).
1. Compute \( f(3) \):
\[
f(3) = 3^2 + 7 = 9 + 7 = 16
\]
2. Now compute \( g(16) \):
\[
g(16) = 16^2 - 4 = 256 - 4 = 252
\]
**Final Answer for Part d:**
\[
(g \circ f)(3) = 252
\]
In summary:
- a: \( (f \circ g)(x) = x^4 - 8x^2 + 23 \)
- b: \( (g \circ f)(x) = x^4 + 14x^2 + 45 \)
- c: \( (f \circ g)(3) = 32 \)
- d: \( (g \circ f)(3) = 252 \)
- \( f(x) = x^2 + 7 \)
- \( g(x) = x^2 - 4 \)
we need to calculate \( (f \circ g)(x) \) and \( (g \circ f)(x) \).
### Part a: \( (f \circ g)(x) \)
The composition \( (f \circ g)(x) \) means we substitute \( g(x) \) into \( f(x) \).
1. Compute \( g(x) = x^2 - 4 \).
2. Substitute \( g(x) \) into \( f \):
\[
f(g(x)) = f(x^2 - 4) = (x^2 - 4)^2 + 7
\]
Now, expand \( (x^2 - 4)^2 \):
\[
(x^2 - 4)^2 = x^4 - 8x^2 + 16
\]
Now substitute back into \( f(g(x)) \):
\[
f(g(x)) = x^4 - 8x^2 + 16 + 7
\]
Simplify:
\[
f(g(x)) = x^4 - 8x^2 + 23
\]
**Final Answer for Part a:**
\[
(f \circ g)(x) = x^4 - 8x^2 + 23
\]
---
### Part b: \( (g \circ f)(x) \)
Now compute \( (g \circ f)(x) \).
1. Compute \( f(x) = x^2 + 7 \).
2. Substitute \( f(x) \) into \( g \):
\[
g(f(x)) = g(x^2 + 7) = (x^2 + 7)^2 - 4
\]
Now, expand \( (x^2 + 7)^2 \):
\[
(x^2 + 7)^2 = x^4 + 14x^2 + 49
\]
Now substitute back into \( g(f(x)) \):
\[
g(f(x)) = x^4 + 14x^2 + 49 - 4
\]
Simplify:
\[
g(f(x)) = x^4 + 14x^2 + 45
\]
**Final Answer for Part b:**
\[
(g \circ f)(x) = x^4 + 14x^2 + 45
\]
---
### Part c: \( (f \circ g)(3) \)
Now compute \( (f \circ g)(3) \).
1. Compute \( g(3) \):
\[
g(3) = 3^2 - 4 = 9 - 4 = 5
\]
2. Now compute \( f(5) \):
\[
f(5) = 5^2 + 7 = 25 + 7 = 32
\]
**Final Answer for Part c:**
\[
(f \circ g)(3) = 32
\]
---
### Part d: \( (g \circ f)(3) \)
Now compute \( (g \circ f)(3) \).
1. Compute \( f(3) \):
\[
f(3) = 3^2 + 7 = 9 + 7 = 16
\]
2. Now compute \( g(16) \):
\[
g(16) = 16^2 - 4 = 256 - 4 = 252
\]
**Final Answer for Part d:**
\[
(g \circ f)(3) = 252
\]
In summary:
- a: \( (f \circ g)(x) = x^4 - 8x^2 + 23 \)
- b: \( (g \circ f)(x) = x^4 + 14x^2 + 45 \)
- c: \( (f \circ g)(3) = 32 \)
- d: \( (g \circ f)(3) = 252 \)