Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
2
![**Finding Composite Functions**
In order to find the rules for the composite functions \( f \circ g \) and \( g \circ f \), we need to understand their definitions and use the provided functions.
1. **Given Functions:**
- \( f(x) = 8x^2 + 9x + 4 \)
- \( g(x) = x + 7 \)
2. **Composite Function \( f \circ g \):**
- To find \( f \circ g \), we need to substitute \( g(x) \) into \( f(x) \).
- \( f \circ g = f(g(x)) = f(x + 7) \).
3. **Composite Function \( g \circ f \):**
- To find \( g \circ f \), we need to substitute \( f(x) \) into \( g(x) \).
- \( g \circ f = g(f(x)) = g(8x^2 + 9x + 4) \).
**Steps to Find \( f \circ g \):**
- Substitute \( x + 7 \) in place of \( x \) in \( f(x) \):
\[
f(x + 7) = 8(x + 7)^2 + 9(x + 7) + 4
\]
- Simplify the expression:
\[
= 8(x^2 + 14x + 49) + 9(x + 7) + 4
\]
\[
= 8x^2 + 112x + 392 + 9x + 63 + 4
\]
\[
= 8x^2 + 121x + 459
\]
**Thus, \( f \circ g = 8x^2 + 121x + 459 \).**
**Steps to Find \( g \circ f \):**
- Substitute \( 8x^2 + 9x + 4 \) in place of \( x \) in \( g(x) \):
\[
g(8x^2 + 9x + 4) = (8x^2 + 9x + 4) + 7
\]
- Simplify the expression:
\[
= 8x^2 + 9](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8eeb6921-a110-4984-a04e-b3e01fc97c1d%2Fe4dd5f72-de71-4a81-80d6-e09a592b4b4f%2Fckke3b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding Composite Functions**
In order to find the rules for the composite functions \( f \circ g \) and \( g \circ f \), we need to understand their definitions and use the provided functions.
1. **Given Functions:**
- \( f(x) = 8x^2 + 9x + 4 \)
- \( g(x) = x + 7 \)
2. **Composite Function \( f \circ g \):**
- To find \( f \circ g \), we need to substitute \( g(x) \) into \( f(x) \).
- \( f \circ g = f(g(x)) = f(x + 7) \).
3. **Composite Function \( g \circ f \):**
- To find \( g \circ f \), we need to substitute \( f(x) \) into \( g(x) \).
- \( g \circ f = g(f(x)) = g(8x^2 + 9x + 4) \).
**Steps to Find \( f \circ g \):**
- Substitute \( x + 7 \) in place of \( x \) in \( f(x) \):
\[
f(x + 7) = 8(x + 7)^2 + 9(x + 7) + 4
\]
- Simplify the expression:
\[
= 8(x^2 + 14x + 49) + 9(x + 7) + 4
\]
\[
= 8x^2 + 112x + 392 + 9x + 63 + 4
\]
\[
= 8x^2 + 121x + 459
\]
**Thus, \( f \circ g = 8x^2 + 121x + 459 \).**
**Steps to Find \( g \circ f \):**
- Substitute \( 8x^2 + 9x + 4 \) in place of \( x \) in \( g(x) \):
\[
g(8x^2 + 9x + 4) = (8x^2 + 9x + 4) + 7
\]
- Simplify the expression:
\[
= 8x^2 + 9
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