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 Statement:**
Given the functions \( f(x) = 3x + 4 \) and \( g(x) = x^2 \), find \( (f \circ g)(x) \) and its domain.
**Solution Steps:**
1. **Understanding Composition of Functions:**
- The notation \( (f \circ g)(x) \) represents the composition of the two functions \( f \) and \( g \), which is defined as \( f(g(x)) \).
2. **Calculate \( (f \circ g)(x) \):**
- Start by finding \( g(x) \).
\[
g(x) = x^2
\]
- Next, substitute \( g(x) \) into \( f \).
\[
f(g(x)) = f(x^2)
\]
- Since \( f(x) = 3x + 4 \), substitute \( x^2 \) for \( x \) in \( f(x) \).
\[
f(x^2) = 3(x^2) + 4 = 3x^2 + 4
\]
- Therefore, \( (f \circ g)(x) = 3x^2 + 4 \).
3. **Determine the Domain of \( (f \circ g)(x) \):**
- The domain of \( (f \circ g)(x) \) is the set of all \( x \) values for which the expression \( 3x^2 + 4 \) is defined.
- Since \( 3x^2 + 4 \) is a polynomial, it is defined for all real numbers (\( \mathbb{R} \)).
**Answer:**
\( (f \circ g)(x) = 3x^2 + 4 \)
The domain of \( (f \circ g)(x) \) is \( \mathbb{R} \) (all real numbers).
---
This explanation provides step-by-step guidance to understanding and solving the function composition problem presented.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba45ef30-7373-40a6-93b1-6069d924f96b%2F0081c22b-1114-41c7-b7bd-359fe76b1425%2Fz0ivfc9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
**Problem Statement:**
Given the functions \( f(x) = 3x + 4 \) and \( g(x) = x^2 \), find \( (f \circ g)(x) \) and its domain.
**Solution Steps:**
1. **Understanding Composition of Functions:**
- The notation \( (f \circ g)(x) \) represents the composition of the two functions \( f \) and \( g \), which is defined as \( f(g(x)) \).
2. **Calculate \( (f \circ g)(x) \):**
- Start by finding \( g(x) \).
\[
g(x) = x^2
\]
- Next, substitute \( g(x) \) into \( f \).
\[
f(g(x)) = f(x^2)
\]
- Since \( f(x) = 3x + 4 \), substitute \( x^2 \) for \( x \) in \( f(x) \).
\[
f(x^2) = 3(x^2) + 4 = 3x^2 + 4
\]
- Therefore, \( (f \circ g)(x) = 3x^2 + 4 \).
3. **Determine the Domain of \( (f \circ g)(x) \):**
- The domain of \( (f \circ g)(x) \) is the set of all \( x \) values for which the expression \( 3x^2 + 4 \) is defined.
- Since \( 3x^2 + 4 \) is a polynomial, it is defined for all real numbers (\( \mathbb{R} \)).
**Answer:**
\( (f \circ g)(x) = 3x^2 + 4 \)
The domain of \( (f \circ g)(x) \) is \( \mathbb{R} \) (all real numbers).
---
This explanation provides step-by-step guidance to understanding and solving the function composition problem presented.
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