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 focus on graphing a piecewise-defined function, specifically the function 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}.

Understanding Piecewise-Defined Functions

A piecewise-defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. The function is typically defined as a set of rules, where each rule applies to a specific interval of the domain. For example, the function 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} is a piecewise-defined function, where the first rule applies to the interval x≤−1x \leq -1 and the second rule applies to the interval x \textgreater −1x \ \textgreater \ -1.

Graphing Piecewise-Defined Functions

To graph a piecewise-defined function, we need to graph each sub-function separately and then combine them to form the final graph. In the case of the function 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}, we need to graph the two sub-functions separately and then combine them to form the final graph.

Graphing the First Sub-Function

The first sub-function is f(x)=x−2f(x) = x - 2 for x≤−1x \leq -1. To graph this sub-function, we can start by finding the x-intercept, which is the point where the graph crosses the x-axis. To find the x-intercept, we can set f(x)=0f(x) = 0 and solve for xx. This gives us x−2=0x - 2 = 0, which implies that x=2x = 2. However, since x≤−1x \leq -1, the x-intercept is actually at x=−1x = -1. We can then use this point to graph the sub-function.

Graphing the Second Sub-Function

The second sub-function is f(x)=−3f(x) = -3 for x \textgreater −1x \ \textgreater \ -1. To graph this sub-function, we can start by finding the y-intercept, which is the point where the graph crosses the y-axis. To find the y-intercept, we can set x=0x = 0 and solve for f(x)f(x). This gives us f(x)=−3f(x) = -3, which implies that the y-intercept is at (0,−3)(0, -3). We can then use this point to graph the sub-function.

Combining the Sub-Functions

To combine the sub-functions, we need to graph each sub-function separately and then combine them to form the final graph. In the case of the function 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}, we can graph the first sub-function for x≤−1x \leq -1 and the second sub-function for x \textgreater −1x \ \textgreater \ -1. The final graph will be a combination of the two sub-functions.

Graphing the Final Function

To graph the final function, we can use the following steps:

  1. Graph the first sub-function for x≤−1x \leq -1.
  2. Graph the second sub-function for x \textgreater −1x \ \textgreater \ -1.
  3. Combine the two sub-functions to form the final graph.

The final graph will be a combination of the two sub-functions, with the first sub-function applying to the interval x≤−1x \leq -1 and the second sub-function applying to the interval x \textgreater −1x \ \textgreater \ -1.

Conclusion

In this article, we have discussed graphing piecewise-defined functions, specifically the function 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}. We have shown how to graph each sub-function separately and then combine them to form the final graph. The final graph will be a combination of the two sub-functions, with the first sub-function applying to the interval x≤−1x \leq -1 and the second sub-function applying to the interval x \textgreater −1x \ \textgreater \ -1.

Choosing the Correct Graph

To choose the correct graph, we need to look at the final graph and determine which one matches the function 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}. The correct graph will be the one that matches the function.

Answer

The correct graph is:

Graph A

This graph matches the function 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}.

Discussion

The graph of a piecewise-defined function can be a combination of multiple sub-functions, each applied to a specific interval of the domain. In the case of the function 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}, the graph is a combination of the two sub-functions, with the first sub-function applying to the interval x≤−1x \leq -1 and the second sub-function applying to the interval x \textgreater −1x \ \textgreater \ -1.

Key Takeaways

  • A piecewise-defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.
  • To graph a piecewise-defined function, we need to graph each sub-function separately and then combine them to form the final graph.
  • The final graph will be a combination of the two sub-functions, with the first sub-function applying to the interval x≤−1x \leq -1 and the second sub-function applying to the interval x \textgreater −1x \ \textgreater \ -1.

References

  • [1] "Piecewise-Defined Functions" by Math Open Reference
  • [2] "Graphing Piecewise-Defined Functions" by Khan Academy
    Graphing Piecewise-Defined Functions: A Q&A Guide =====================================================

Introduction

In our previous article, we discussed graphing piecewise-defined functions, specifically the function 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}. In this article, we will provide a Q&A guide to help you better understand graphing piecewise-defined functions.

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 graph a piecewise-defined function?

A: To graph a piecewise-defined function, you need to graph each sub-function separately and then combine them to form the final graph.

Q: What are the steps to graph a piecewise-defined function?

A: The steps to graph a piecewise-defined function are:

  1. Graph the first sub-function for the specified interval.
  2. Graph the second sub-function for the specified interval.
  3. Combine the two sub-functions to form the final graph.

Q: How do I determine which sub-function to graph first?

A: To determine which sub-function to graph first, you need to look at the intervals specified in the piecewise-defined function. The first sub-function should be graphed for the interval that comes first in the piecewise-defined function.

Q: What if the intervals overlap?

A: If the intervals overlap, you need to graph the sub-functions separately and then combine them to form the final graph. The sub-function that comes first in the piecewise-defined function should be graphed for the overlapping interval.

Q: How do I graph a piecewise-defined function with multiple sub-functions?

A: To graph a piecewise-defined function with multiple sub-functions, you need to graph each sub-function separately and then combine them to form the final graph. The sub-functions should be graphed in the order specified in the piecewise-defined function.

Q: What if I'm having trouble graphing a piecewise-defined function?

A: If you're having trouble graphing a piecewise-defined function, try breaking it down into smaller steps. Graph each sub-function separately and then combine them to form the final graph. If you're still having trouble, try using a graphing calculator or software to help you visualize the function.

Q: Can I graph a piecewise-defined function using a graphing calculator or software?

A: Yes, you can graph a piecewise-defined function using a graphing calculator or software. Most graphing calculators and software programs have built-in functions for graphing piecewise-defined functions.

Q: What are some common mistakes to avoid when graphing piecewise-defined functions?

A: Some common mistakes to avoid when graphing piecewise-defined functions include:

  • Graphing the wrong sub-function for the specified interval.
  • Failing to combine the sub-functions correctly.
  • Not using the correct notation for the piecewise-defined function.

Conclusion

Graphing piecewise-defined functions can be a challenging task, but with practice and patience, you can become proficient in graphing these types of functions. Remember to break down the function into smaller steps, graph each sub-function separately, and then combine them to form the final graph. If you're still having trouble, try using a graphing calculator or software to help you visualize the function.

Key Takeaways

  • A piecewise-defined function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.
  • To graph a piecewise-defined function, you need to graph each sub-function separately and then combine them to form the final graph.
  • The sub-functions should be graphed in the order specified in the piecewise-defined function.
  • You can graph a piecewise-defined function using a graphing calculator or software.

References

  • [1] "Piecewise-Defined Functions" by Math Open Reference
  • [2] "Graphing Piecewise-Defined Functions" by Khan Academy