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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![**Mathematics Function Composition: Problem Solving**
**Given Functions:**
Let \( f(x) = 2x - 1 \) and \( g(x) = x^2 - 2x + 1 \), for \( x \in \mathbb{R} \).
**Tasks:**
a) Find \((g \circ f)(x)\).
b) Determine the Domain and Range of \((f \circ g)(x)\).
**Solution Explanation:**
a) To find \((g \circ f)(x)\), we substitute \( f(x) \) into \( g(x) \).
- **Step 1:** Calculate \( f(x) = 2x - 1 \).
- **Step 2:** Substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(2x - 1) = (2x - 1)^2 - 2(2x - 1) + 1
\]
- Expand \((2x - 1)^2\):
\[
(2x - 1)^2 = 4x^2 - 4x + 1
\]
- Calculate the linear terms:
\[
-2(2x - 1) = -4x + 2
\]
- Substitute back into the expression:
\[
g(f(x)) = 4x^2 - 4x + 1 - 4x + 2 + 1 = 4x^2 - 8x + 4
\]
b) To determine the domain and range of \((f \circ g)(x)\), consider:
- **Domain:** \(\mathbb{R}\), as polynomial functions are defined for all real numbers.
- **Range:** The output of \((f \circ g)(x)\) based on specific calculations and behaviors of quadratic functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8beeba18-127d-4f79-b572-133114d91afd%2Fd0de1407-273b-4a3d-a891-017ef5a48215%2Fzxfuuks_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematics Function Composition: Problem Solving**
**Given Functions:**
Let \( f(x) = 2x - 1 \) and \( g(x) = x^2 - 2x + 1 \), for \( x \in \mathbb{R} \).
**Tasks:**
a) Find \((g \circ f)(x)\).
b) Determine the Domain and Range of \((f \circ g)(x)\).
**Solution Explanation:**
a) To find \((g \circ f)(x)\), we substitute \( f(x) \) into \( g(x) \).
- **Step 1:** Calculate \( f(x) = 2x - 1 \).
- **Step 2:** Substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(2x - 1) = (2x - 1)^2 - 2(2x - 1) + 1
\]
- Expand \((2x - 1)^2\):
\[
(2x - 1)^2 = 4x^2 - 4x + 1
\]
- Calculate the linear terms:
\[
-2(2x - 1) = -4x + 2
\]
- Substitute back into the expression:
\[
g(f(x)) = 4x^2 - 4x + 1 - 4x + 2 + 1 = 4x^2 - 8x + 4
\]
b) To determine the domain and range of \((f \circ g)(x)\), consider:
- **Domain:** \(\mathbb{R}\), as polynomial functions are defined for all real numbers.
- **Range:** The output of \((f \circ g)(x)\) based on specific calculations and behaviors of quadratic functions.
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