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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![**Function Continuity and Discontinuity Analysis**
### Graph Analysis
**Left Graph:**
- **Axes Description:** The x-axis ranges from -10 to 10, and the y-axis ranges from -10 to 10.
- **Graph Description:** The graph is a piecewise function. It is continuous and parabolic on either side of a point at x = 0 but exhibits a discontinuity at x = 0. The curve dips sharply towards a point at the origin (x=0, y=0) but continues from just below the x-axis on the positive side.
**Right Graph:**
- **Axes Description:** The x-axis ranges from -10 to 10, and the y-axis ranges from -10 to 10.
- **Graph Description:** Similar to the left graph, this is another piecewise function showing continuity elsewhere but a discontinuity at x = 0. The curve approaches the origin but does not connect through it smoothly, indicating the presence of a discontinuity.
### Questions & Solutions
1. Describe the interval(s) on which the function is continuous. (Enter your answer using interval notation.)
[Input Box]
2. If there are any discontinuities, determine whether they are removable. (If an answer does not exist, enter DNE.)
- Removable Discontinuities: \( x = \) [Input Box]
- Nonremovable Discontinuities: \( x = \) [Input Box]
*Submit Answer*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fea03bc01-0c8f-4ac0-bde7-e545ccac1fc8%2Fc86c7f5d-b4b5-4cda-bd55-8e7944c142cf%2F1l505r.jpeg&w=3840&q=75)
Transcribed Image Text:**Function Continuity and Discontinuity Analysis**
### Graph Analysis
**Left Graph:**
- **Axes Description:** The x-axis ranges from -10 to 10, and the y-axis ranges from -10 to 10.
- **Graph Description:** The graph is a piecewise function. It is continuous and parabolic on either side of a point at x = 0 but exhibits a discontinuity at x = 0. The curve dips sharply towards a point at the origin (x=0, y=0) but continues from just below the x-axis on the positive side.
**Right Graph:**
- **Axes Description:** The x-axis ranges from -10 to 10, and the y-axis ranges from -10 to 10.
- **Graph Description:** Similar to the left graph, this is another piecewise function showing continuity elsewhere but a discontinuity at x = 0. The curve approaches the origin but does not connect through it smoothly, indicating the presence of a discontinuity.
### Questions & Solutions
1. Describe the interval(s) on which the function is continuous. (Enter your answer using interval notation.)
[Input Box]
2. If there are any discontinuities, determine whether they are removable. (If an answer does not exist, enter DNE.)
- Removable Discontinuities: \( x = \) [Input Box]
- Nonremovable Discontinuities: \( x = \) [Input Box]
*Submit Answer*
![### Consider the following:
Given function:
\[
f(x) =
\begin{cases}
x^2 - 1, & \text{if } x \leq 0 \\
5x + 1, & \text{if } x > 0
\end{cases}
\]
#### Objective:
Sketch the graph of the function.
#### Graph Explanation:
In the graph provided, there are two distinct parts described by different equations based on the value of \(x\).
1. **For \(x \leq 0\):**
- The function is \(f(x) = x^2 - 1\).
- This represents a parabola opening upwards, shifted downward by 1 unit.
2. **For \(x > 0\):**
- The function changes to \(f(x) = 5x + 1\).
- This represents a linear graph with a slope of 5 and a y-intercept at 1.
#### Detailed Graph Plot:
- **Left Side (\(x \leq 0\))**:
- The parabola touches the y-axis at \((0, -1)\) and extends infinitely upwards as \(x\) becomes more negative.
- Example points:
- \( (-2, 3) \)
- \( (-1, 0) \)
- \( (0, -1) \)
- **Right Side (\(x > 0\))**:
- The line starts at \((0^+, 1)\) (meaning just to the right of \((0, 1)\)) and extends infinitely upwards as \(x\) increases.
- Example points:
- \( (1, 6) \)
- \( (2, 11) \)
- \( (3, 16) \)
#### Key Points:
- At \(x = 0\), observe the discontinuity:
- The left part of the graph ends at \((0, -1)\).
- The right part starts just right of \((0, 1)\).
In the graphs, the points where the function changes and the continuous sections are carefully illustrated. Each segment accurately depicts the corresponding equation for the given domain of \(x\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fea03bc01-0c8f-4ac0-bde7-e545ccac1fc8%2Fc86c7f5d-b4b5-4cda-bd55-8e7944c142cf%2Fks3cshls.jpeg&w=3840&q=75)
Transcribed Image Text:### Consider the following:
Given function:
\[
f(x) =
\begin{cases}
x^2 - 1, & \text{if } x \leq 0 \\
5x + 1, & \text{if } x > 0
\end{cases}
\]
#### Objective:
Sketch the graph of the function.
#### Graph Explanation:
In the graph provided, there are two distinct parts described by different equations based on the value of \(x\).
1. **For \(x \leq 0\):**
- The function is \(f(x) = x^2 - 1\).
- This represents a parabola opening upwards, shifted downward by 1 unit.
2. **For \(x > 0\):**
- The function changes to \(f(x) = 5x + 1\).
- This represents a linear graph with a slope of 5 and a y-intercept at 1.
#### Detailed Graph Plot:
- **Left Side (\(x \leq 0\))**:
- The parabola touches the y-axis at \((0, -1)\) and extends infinitely upwards as \(x\) becomes more negative.
- Example points:
- \( (-2, 3) \)
- \( (-1, 0) \)
- \( (0, -1) \)
- **Right Side (\(x > 0\))**:
- The line starts at \((0^+, 1)\) (meaning just to the right of \((0, 1)\)) and extends infinitely upwards as \(x\) increases.
- Example points:
- \( (1, 6) \)
- \( (2, 11) \)
- \( (3, 16) \)
#### Key Points:
- At \(x = 0\), observe the discontinuity:
- The left part of the graph ends at \((0, -1)\).
- The right part starts just right of \((0, 1)\).
In the graphs, the points where the function changes and the continuous sections are carefully illustrated. Each segment accurately depicts the corresponding equation for the given domain of \(x\).
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