**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 & Solutions1. 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 $x$.

Use online graphing calculator and sketch the graph of the function as follows. From the above…

Answered: Consider the following. x² - 1, f(x) 5x…

Math

Calculus

Consider the following. x² - 1, f(x) 5x + 1, x > 0 Sketch the graph of the function. 10H

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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Question

**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 $x$.

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