Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Problem 7:**
For the functions \( f(x) = 3x^2 - 5x + 7 \) and \( g(x) = 4x + 8 \), find \( (f \circ g)(x) \).
**Solution Explanation:**
1. **Understanding Composition of Functions:**
The notation \( (f \circ g)(x) \) refers to the composition of the functions \( f \) and \( g \). This means you substitute \( g(x) \) into \( f(x) \).
2. **Substituting \( g(x) \) into \( f(x) \):**
To find \( (f \circ g)(x) \), replace every occurrence of \( x \) in \( f(x) \) with \( g(x) \).
So, \( f(g(x)) = f(4x + 8) \).
3. **Calculating \( f(4x + 8) \):**
Substitute \( 4x + 8 \) into the expression for \( f(x) \):
\[
f(4x + 8) = 3(4x + 8)^2 - 5(4x + 8) + 7
\]
- **Expanding \( (4x + 8)^2 \):**
\[
(4x + 8)^2 = 16x^2 + 64x + 64
\]
- **Substituting back into \( f(x) \):**
\[
f(4x + 8) = 3(16x^2 + 64x + 64) - 5(4x + 8) + 7
\]
- **Distribute the coefficients:**
\[
= 48x^2 + 192x + 192 - 20x - 40 + 7
\]
- **Combine like terms:**
\[
= 48x^2 + 172x + 159
\]
4. **Final Result:**
The expression for \( (f \circ g)(x) \) is \( 48x^2 + 172x + 159 \).
This result provides the function that represents the composition of \( f \) and \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70a8cbee-10b6-427b-ac1c-41a66a493bbd%2Fa93c4794-0a2b-4060-9300-9b5543dc3703%2Fyjc7ov_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 7:**
For the functions \( f(x) = 3x^2 - 5x + 7 \) and \( g(x) = 4x + 8 \), find \( (f \circ g)(x) \).
**Solution Explanation:**
1. **Understanding Composition of Functions:**
The notation \( (f \circ g)(x) \) refers to the composition of the functions \( f \) and \( g \). This means you substitute \( g(x) \) into \( f(x) \).
2. **Substituting \( g(x) \) into \( f(x) \):**
To find \( (f \circ g)(x) \), replace every occurrence of \( x \) in \( f(x) \) with \( g(x) \).
So, \( f(g(x)) = f(4x + 8) \).
3. **Calculating \( f(4x + 8) \):**
Substitute \( 4x + 8 \) into the expression for \( f(x) \):
\[
f(4x + 8) = 3(4x + 8)^2 - 5(4x + 8) + 7
\]
- **Expanding \( (4x + 8)^2 \):**
\[
(4x + 8)^2 = 16x^2 + 64x + 64
\]
- **Substituting back into \( f(x) \):**
\[
f(4x + 8) = 3(16x^2 + 64x + 64) - 5(4x + 8) + 7
\]
- **Distribute the coefficients:**
\[
= 48x^2 + 192x + 192 - 20x - 40 + 7
\]
- **Combine like terms:**
\[
= 48x^2 + 172x + 159
\]
4. **Final Result:**
The expression for \( (f \circ g)(x) \) is \( 48x^2 + 172x + 159 \).
This result provides the function that represents the composition of \( f \) and \(
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