Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
ChapterP: Prerequisites
SectionP.7: A Library Of Parent Functions
Problem 47E
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![The given image presents a mathematical problem involving function composition. Here's the text transcribed and explained:
---
Let \( f(x) = 2x + 5 \) and \( g(x) = 3x^2 + 3x \).
After simplifying,
\[ (f \circ g)(x) = \]
---
To interpret this problem in the context of an educational website:
### Topic: Function Composition
#### Problem Description:
In this exercise, you are given two functions:
- \( f(x) = 2x + 5 \)
- \( g(x) = 3x^2 + 3x \)
You are tasked with finding the composition of these two functions, denoted as \( (f \circ g)(x) \).
#### Steps to Solve:
1. **Understand the Notation**: The composition \( (f \circ g)(x) \) means you need to apply \( g(x) \) first and then apply \( f \) to the result of \( g(x) \).
2. **Substitute \( g(x) \) in \( f \)**:
- Start by substituting \( g(x) = 3x^2 + 3x \) into \( f \).
- Thus, \( (f \circ g)(x) = f(g(x)) = f(3x^2 + 3x) \).
3. **Apply the Function \( f \)**:
- \( f(t) = 2t + 5 \) where \( t = g(x) \).
- Replace \( t \) with \( 3x^2 + 3x \).
- So, \( f(3x^2 + 3x) = 2(3x^2 + 3x) + 5 \).
4. **Simplify**:
- Distribute the 2: \( 2(3x^2 + 3x) = 6x^2 + 6x \).
- Add the constant term: \( 6x^2 + 6x + 5 \).
Therefore, after simplifying, the composition \( (f \circ g)(x) \) is:
\[ (f \circ g)(x) = 6x^2 + 6x + 5 \]
#### Conclusion:
The composed function \( (f \circ g)(x) \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa78a88d7-60f0-4b9e-a272-6ff852667c68%2Fb9a0bfb8-0957-4b33-ad41-67c8279df270%2Flhx28b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The given image presents a mathematical problem involving function composition. Here's the text transcribed and explained:
---
Let \( f(x) = 2x + 5 \) and \( g(x) = 3x^2 + 3x \).
After simplifying,
\[ (f \circ g)(x) = \]
---
To interpret this problem in the context of an educational website:
### Topic: Function Composition
#### Problem Description:
In this exercise, you are given two functions:
- \( f(x) = 2x + 5 \)
- \( g(x) = 3x^2 + 3x \)
You are tasked with finding the composition of these two functions, denoted as \( (f \circ g)(x) \).
#### Steps to Solve:
1. **Understand the Notation**: The composition \( (f \circ g)(x) \) means you need to apply \( g(x) \) first and then apply \( f \) to the result of \( g(x) \).
2. **Substitute \( g(x) \) in \( f \)**:
- Start by substituting \( g(x) = 3x^2 + 3x \) into \( f \).
- Thus, \( (f \circ g)(x) = f(g(x)) = f(3x^2 + 3x) \).
3. **Apply the Function \( f \)**:
- \( f(t) = 2t + 5 \) where \( t = g(x) \).
- Replace \( t \) with \( 3x^2 + 3x \).
- So, \( f(3x^2 + 3x) = 2(3x^2 + 3x) + 5 \).
4. **Simplify**:
- Distribute the 2: \( 2(3x^2 + 3x) = 6x^2 + 6x \).
- Add the constant term: \( 6x^2 + 6x + 5 \).
Therefore, after simplifying, the composition \( (f \circ g)(x) \) is:
\[ (f \circ g)(x) = 6x^2 + 6x + 5 \]
#### Conclusion:
The composed function \( (f \circ g)(x) \
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