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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![### Problem 16
Sketch the graph of the function \( f(x) \) defined as:
\[
f(x) =
\begin{cases}
3x + 7 & \text{if } x \leq -3 \\
4x^2 - 1 & \text{if } x > -3
\end{cases}
\]
### Explanation
This is a piecewise function with two different expressions:
1. **For \( x \leq -3 \):**
- The function is \( 3x + 7 \).
- This is a linear equation, which forms a straight line on the graph. The slope is 3, and the y-intercept is 7.
2. **For \( x > -3 \):**
- The function is \( 4x^2 - 1 \).
- This is a quadratic equation, which forms a parabola on the graph. The parabola opens upwards since the coefficient of \( x^2 \) is positive (4). The vertex of this parabola needs to be determined by further calculation or graphing.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F63bbee30-bb30-4176-a66e-fefcba8e1595%2F9005c2ab-92bf-4047-b510-fae4372be607%2Fvsef2sm_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 16
Sketch the graph of the function \( f(x) \) defined as:
\[
f(x) =
\begin{cases}
3x + 7 & \text{if } x \leq -3 \\
4x^2 - 1 & \text{if } x > -3
\end{cases}
\]
### Explanation
This is a piecewise function with two different expressions:
1. **For \( x \leq -3 \):**
- The function is \( 3x + 7 \).
- This is a linear equation, which forms a straight line on the graph. The slope is 3, and the y-intercept is 7.
2. **For \( x > -3 \):**
- The function is \( 4x^2 - 1 \).
- This is a quadratic equation, which forms a parabola on the graph. The parabola opens upwards since the coefficient of \( x^2 \) is positive (4). The vertex of this parabola needs to be determined by further calculation or graphing.
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