Graph The Piecewise-defined Function:$ f(x) = \begin{cases} x - 2 & \text{if } X \leq -1 \\ -3 & \text{if } X \ \textgreater \ -1 \end{cases} $Choose The Correct Graph.A. B. C. D.

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Introduction

In mathematics, a piecewise-defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. These functions are commonly used to model real-world phenomena that exhibit different behaviors in different regions. In this article, we will explore how to graph a piecewise-defined function, using the given function as an example.

Understanding the Function

The given function is defined as:

f(x)={x−2if x≤−1−3if x \textgreater −1f(x) = \begin{cases} x - 2 & \text{if } x \leq -1 \\ -3 & \text{if } x \ \textgreater \ -1 \end{cases}

This function has two sub-functions:

  1. For x≤−1x \leq -1, the function is defined as f(x)=x−2f(x) = x - 2.
  2. For x \textgreater −1x \ \textgreater \ -1, the function is defined as f(x)=−3f(x) = -3.

Graphing the Function

To graph the function, we need to graph each sub-function separately and then combine them.

Graphing the First Sub-Function

The first sub-function is defined as f(x)=x−2f(x) = x - 2 for x≤−1x \leq -1. This is a linear function with a slope of 1 and a y-intercept of -2. To graph this function, we can use the slope-intercept form of a linear equation:

y=mx+by = mx + b

where mm is the slope and bb is the y-intercept. In this case, m=1m = 1 and b=−2b = -2.

**Graph of the First Sub-Function**
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*   **Domain:** $x \leq -1$
*   **Range:** $y \leq -1$
*   **Slope:** 1
*   **Y-intercept:** -2

Graphing the Second Sub-Function

The second sub-function is defined as f(x)=−3f(x) = -3 for x \textgreater −1x \ \textgreater \ -1. This is a constant function with a value of -3. To graph this function, we can simply draw a horizontal line at y=−3y = -3.

**Graph of the Second Sub-Function**
---------------------------------

*   **Domain:** $x \  \textgreater \  -1$
*   **Range:** $y = -3$

Combining the Sub-Functions

To graph the entire function, we need to combine the two sub-functions. We can do this by drawing the graph of the first sub-function for x≤−1x \leq -1 and the graph of the second sub-function for x \textgreater −1x \ \textgreater \ -1.

**Graph of the Piecewise-Defined Function**
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*   **Domain:** $x \leq -1$ or $x \  \textgreater \  -1$
*   **Range:** $y \leq -1$ or $y = -3$

Choosing the Correct Graph

Now that we have graphed the piecewise-defined function, we can choose the correct graph from the options provided.

**Discussion**
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*   **Graph A:** This graph shows a linear function with a slope of 1 and a y-intercept of -2 for $x \leq -1$, and a constant function with a value of -3 for $x \  \textgreater \  -1$.
*   **Graph B:** This graph shows a linear function with a slope of 1 and a y-intercept of -2 for $x \  \textgreater \  -1$, and a constant function with a value of -3 for $x \leq -1$.
*   **Graph C:** This graph shows a linear function with a slope of 1 and a y-intercept of -2 for $x \leq -1$, and a constant function with a value of -3 for $x \  \textgreater \  -1$.
*   **Graph D:** This graph shows a linear function with a slope of 1 and a y-intercept of -2 for $x \  \textgreater \  -1$, and a constant function with a value of -3 for $x \leq -1$.

The correct graph is **Graph C**.

Conclusion

Introduction

In our previous article, we explored how to graph a piecewise-defined function using the given function as an example. In this article, we will answer some common questions that students often have when it comes to graphing piecewise-defined functions.

Q&A

Q: What is a piecewise-defined function?

A: A piecewise-defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.

Q: How do I know which sub-function to use for a given value of x?

A: To determine which sub-function to use, you need to check the value of x and see which interval it falls into. If x is less than or equal to -1, use the first sub-function. If x is greater than -1, use the second sub-function.

Q: Can I have more than two sub-functions?

A: Yes, you can have more than two sub-functions. For example, you could have a function defined as:

f(x)={x−2if x≤−1−3if −1<x≤2x+1if x>2f(x) = \begin{cases} x - 2 & \text{if } x \leq -1 \\ -3 & \text{if } -1 < x \leq 2 \\ x + 1 & \text{if } x > 2 \end{cases}

Q: How do I graph a piecewise-defined function with more than two sub-functions?

A: To graph a piecewise-defined function with more than two sub-functions, you need to graph each sub-function separately and then combine them. You can use the same steps as before, but you will need to add more sub-functions to the graph.

Q: Can I have a piecewise-defined function with no sub-functions?

A: No, a piecewise-defined function must have at least two sub-functions. If you have a function that is defined by only one sub-function, it is not a piecewise-defined function.

Q: How do I choose the correct graph from the options provided?

A: To choose the correct graph, you need to look at the graph and see which one matches the piecewise-defined function you are graphing. You can use the following steps to help you choose the correct graph:

  1. Check the domain of the function and see which interval each sub-function is defined for.
  2. Check the range of the function and see which value each sub-function takes.
  3. Look at the graph and see which one matches the piecewise-defined function you are graphing.

Q: What if I am not sure which graph is correct?

A: If you are not sure which graph is correct, you can try graphing the function yourself and see which graph matches the one you graphed. You can also ask your teacher or a tutor for help.

Conclusion

In this article, we have answered some common questions that students often have when it comes to graphing piecewise-defined functions. We have also provided some tips and tricks to help you choose the correct graph from the options provided. By following the steps outlined in this article, you should be able to graph any piecewise-defined function with confidence.

Additional Resources

  • Graphing Piecewise-Defined Functions: This article provides a step-by-step guide to graphing piecewise-defined functions.
  • Piecewise-Defined Functions: This article provides a detailed explanation of piecewise-defined functions and how to graph them.
  • Graphing Piecewise-Defined Functions Practice: This article provides some practice problems to help you graph piecewise-defined functions.

Common Mistakes

  • Not checking the domain and range of the function: Make sure to check the domain and range of the function before graphing it.
  • Not using the correct sub-function: Make sure to use the correct sub-function for each interval of the domain.
  • Not graphing the function correctly: Make sure to graph the function correctly, including the domain and range.

Conclusion

Graphing piecewise-defined functions can be a challenging task, but with practice and patience, you can become proficient in graphing these functions. Remember to check the domain and range of the function, use the correct sub-function for each interval of the domain, and graph the function correctly. By following these steps, you should be able to graph any piecewise-defined function with confidence.