**Graph of a Function and Its Symmetry**The graph of a function defined for $x \geq 0$ is given in the first diagram. ### Part (a)**Complete the graph for $x < 0$ to make an even function.**An even function satisfies the property $f(x) = f(-x)$. This means the graph must be symmetric with respect to the y-axis. Below the given function graph, you have four possible graphs that attempt to complete the given function for $x < 0$:1. **Left Side Graph 1:** - This graph correctly mirrors the original function on the given side over the y-axis.2. **Right Side Graph 1:** - This graph includes mirrored portions, but not all parts correctly overlay with the original function.3. **Left Side Graph 2:** - This graph, again, mirrors the original but doesn’t accurately reflect the peaks and troughs of the given function over the y-axis.4. **Right Side Graph 2:** - Similarly, this graph doesn’t accurately reflect the symmetry required for an even function.### Part (b)**Complete the graph for $x < 0$ to make an odd function.**An odd function satisfies the property $f(-x) = -f(x)$. This means the graph must be symmetric about the origin. Below are four additional possible ways the function can be extended to $x < 0$:1. **Left Side Graph 1:** - The graph correctly reflects the original function rotated 180 degrees about the origin.2. **Right Side Graph 1:** - This completion also shows an appropriate rotation indicating an odd function.3. **Left Side Graph 2:** - This graph reflects incorrect symmetry, which doesn’t satisfy the property of an odd function.4. **Right Side Graph 2:** - This graph shows an inaccurate rotation and doesn’t meet the requirements for an odd function. ### Detailed Explanation of Diagrams- **Given Graph:** Displays a function from $x = 0$ to $x > 0$.- **Even Function Completion Options:** - The correct diagrams provide a y-axis reflection of the given function ensuring symmetry.- **Odd Function Completion Options:** - The accurate diagrams show a 180-degree rotation ensuring the function satisfies the origin

There graph of an even function is symmetric about y axis.

Answered: The graph of a function defined for…

The graph of a function defined for x20is given. y (2) Complete the graph forx< 0 to make an even function. vinain (b) Complete the graph for x<0 to make an odd function. y

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

**Graph of a Function and Its Symmetry**

The graph of a function defined for $x \geq 0$ is given in the first diagram.

### Part (a)
**Complete the graph for $x < 0$ to make an even function.**

An even function satisfies the property $f(x) = f(-x)$. This means the graph must be symmetric with respect to the y-axis. Below the given function graph, you have four possible graphs that attempt to complete the given function for $x < 0$:

1. **Left Side Graph 1:**
- This graph correctly mirrors the original function on the given side over the y-axis.

2. **Right Side Graph 1:**
- This graph includes mirrored portions, but not all parts correctly overlay with the original function.

3. **Left Side Graph 2:**
- This graph, again, mirrors the original but doesn’t accurately reflect the peaks and troughs of the given function over the y-axis.

4. **Right Side Graph 2:**
- Similarly, this graph doesn’t accurately reflect the symmetry required for an even function.

### Part (b)
**Complete the graph for $x < 0$ to make an odd function.**

An odd function satisfies the property $f(-x) = -f(x)$. This means the graph must be symmetric about the origin. Below are four additional possible ways the function can be extended to $x < 0$:

1. **Left Side Graph 1:**
- The graph correctly reflects the original function rotated 180 degrees about the origin.

2. **Right Side Graph 1:**
- This completion also shows an appropriate rotation indicating an odd function.

3. **Left Side Graph 2:**
- This graph reflects incorrect symmetry, which doesn’t satisfy the property of an odd function.

4. **Right Side Graph 2:**
- This graph shows an inaccurate rotation and doesn’t meet the requirements for an odd function.

### Detailed Explanation of Diagrams

- **Given Graph:** Displays a function from $x = 0$ to $x > 0$.
- **Even Function Completion Options:**
- The correct diagrams provide a y-axis reflection of the given function ensuring symmetry.
- **Odd Function Completion Options:**
- The accurate diagrams show a 180-degree rotation ensuring the function satisfies the origin

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