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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![**Piecewise-Defined Function Analysis**
**Function Description:**
Sketch the graph of the piecewise-defined function. Write the domain and range of the function.
\[
f(x) =
\begin{cases}
|x|, & x \leq 0 \\
4x^2, & x > 0
\end{cases}
\]
**Explanation and Analysis:**
1. **Graph of \( f(x) = |x| \) for \( x \leq 0 \):**
- This part of the function represents the absolute value of \( x \), which is linear for non-positive values. It has a V-shape with the vertex at the origin (0,0) and falls along the line \( y = -x \) for \( x \leq 0 \).
2. **Graph of \( f(x) = 4x^2 \) for \( x > 0 \):**
- This part of the function represents a parabola opening upwards with vertex at the origin (0,0). The function grows rapidly because of the coefficient 4, making it steeper than the basic \( x^2 \) parabola.
**Domain and Range:**
- **Domain:** The function is defined for all real numbers, thus the domain is \( (-\infty, \infty) \).
- **Range:** The range is all non-negative real numbers, as both pieces of the function yield non-negative outputs. Therefore, the range is \( [0, \infty) \).
This piecewise function combines a linear element and a quadratic element on different intervals of the x-axis, creating a distinct shape where continuity at \( x = 0 \) can be observed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc8f23c41-b0a2-4b7f-b455-de9ffcd4a71a%2F9e9aca22-a832-41fe-9828-319f4e1cea09%2Fzlm8f2n_processed.png&w=3840&q=75)
Transcribed Image Text:**Piecewise-Defined Function Analysis**
**Function Description:**
Sketch the graph of the piecewise-defined function. Write the domain and range of the function.
\[
f(x) =
\begin{cases}
|x|, & x \leq 0 \\
4x^2, & x > 0
\end{cases}
\]
**Explanation and Analysis:**
1. **Graph of \( f(x) = |x| \) for \( x \leq 0 \):**
- This part of the function represents the absolute value of \( x \), which is linear for non-positive values. It has a V-shape with the vertex at the origin (0,0) and falls along the line \( y = -x \) for \( x \leq 0 \).
2. **Graph of \( f(x) = 4x^2 \) for \( x > 0 \):**
- This part of the function represents a parabola opening upwards with vertex at the origin (0,0). The function grows rapidly because of the coefficient 4, making it steeper than the basic \( x^2 \) parabola.
**Domain and Range:**
- **Domain:** The function is defined for all real numbers, thus the domain is \( (-\infty, \infty) \).
- **Range:** The range is all non-negative real numbers, as both pieces of the function yield non-negative outputs. Therefore, the range is \( [0, \infty) \).
This piecewise function combines a linear element and a quadratic element on different intervals of the x-axis, creating a distinct shape where continuity at \( x = 0 \) can be observed.
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