Sketch A Graph Of $\[ f(x)=\left\{ \begin{array}{ll} x+7 & \text{if } X \leq -1 \\ 6 & \text{if } -1 \ \textless \ X \leq 2 \\ 0.5x-7 & \text{if } X \ \textgreater \ 2 \end{array} \right. \\]

=====================================================

Introduction

---

In mathematics, a piecewise function is a function that is defined by multiple sub-functions, each of which applies to a specific interval of the domain. These sub-functions are often referred to as "pieces" of the function, and they are combined to form a single function. In this article, we will explore how to sketch the graph of a piecewise function, using the function $f(x)$ as an example.

The Piecewise Function

---

The piecewise function $f(x)$ is defined as:

$$f(x)=\left\{ \begin{array}{ll} x+7 & \text{if } x \leq -1 \\ 6 & \text{if } -1 \ \textless \ x \leq 2 \\ 0.5x-7 & \text{if } x \ \textgreater \ 2 \end{array} \right.$$

This function has three sub-functions, each of which applies to a specific interval of the domain. The first sub-function, $x+7$, applies when $x$ is less than or equal to $-1$. The second sub-function, $6$, applies when $x$ is greater than $-1$ but less than or equal to $2$. The third sub-function, $0.5x-7$, applies when $x$ is greater than $2$.

Sketching the Graph

---

To sketch the graph of the piecewise function, we need to consider each sub-function separately. We will start by sketching the graph of the first sub-function, $x+7$, which applies when $x$ is less than or equal to $-1$.

First Sub-Function: $x+7$

---

The graph of the first sub-function, $x+7$, is a straight line with a slope of $1$ and a y-intercept of $7$. Since this sub-function applies when $x$ is less than or equal to $-1$, we will only consider the part of the graph that lies to the left of $x=-1$.

Second Sub-Function: $6$

---

The graph of the second sub-function, $6$, is a horizontal line that lies at a height of $6$ on the y-axis. Since this sub-function applies when $x$ is greater than $-1$ but less than or equal to $2$, we will only consider the part of the graph that lies between $x=-1$ and $x=2$.

Third Sub-Function: $0.5x-7$

---

The graph of the third sub-function, $0.5x-7$, is a straight line with a slope of $0.5$ and a y-intercept of $-7$. Since this sub-function applies when $x$ is greater than $2$, we will only consider the part of the graph that lies to the right of $x=2$.

Combining the Sub-Functions

---

Now that we have sketched the graph of each sub-function, we can combine them to form the graph of the piecewise function. We will do this by connecting the different parts of the graph at the points where the sub-functions change.

Connecting the Sub-Functions

---

To connect the sub-functions, we need to find the points where the sub-functions change. These points are $x=-1$ and $x=2$. We will connect the sub-functions at these points by drawing a line segment between the two points.

The Final Graph

---

The final graph of the piecewise function is a combination of the three sub-functions, connected at the points where the sub-functions change. The graph has three distinct parts: a straight line with a slope of $1$ and a y-intercept of $7$ for $x$ less than or equal to $-1$, a horizontal line at a height of $6$ for $-1$ less than $x$ less than or equal to $2$, and a straight line with a slope of $0.5$ and a y-intercept of $-7$ for $x$ greater than $2$.

Conclusion

---

Sketching the graph of a piecewise function requires considering each sub-function separately and combining them to form a single graph. By following the steps outlined in this article, you can sketch the graph of a piecewise function and gain a deeper understanding of how these functions work.

Example Use Cases

---

Piecewise functions have many real-world applications, including:

• Modeling real-world phenomena: Piecewise functions can be used to model real-world phenomena that have different behaviors in different intervals.
• Solving optimization problems: Piecewise functions can be used to solve optimization problems that involve different objective functions in different intervals.
• Analyzing data: Piecewise functions can be used to analyze data that has different patterns in different intervals.

Tips and Tricks

---

When sketching the graph of a piecewise function, keep the following tips and tricks in mind:

• Start with the first sub-function: Begin by sketching the graph of the first sub-function, and then add the other sub-functions as needed.
• Use different colors: Use different colors to distinguish between the different sub-functions.
• Label the graph: Label the graph with the function and the intervals where each sub-function applies.

Conclusion

---

Sketching the graph of a piecewise function is a useful skill that can be applied to a wide range of real-world problems. By following the steps outlined in this article, you can gain a deeper understanding of how piecewise functions work and develop the skills you need to sketch the graph of a piecewise function.

=====================================================

Q: What is a piecewise function?

---

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

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

---

A: To determine which sub-function to use, you need to check the value of the input variable (x) and see which interval it falls into. Then, you can use the corresponding sub-function to evaluate the function.

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

---

A: Yes, you can have more than three sub-functions. In fact, you can have any number of sub-functions, as long as they are defined for different intervals of the domain.

Q: How do I sketch the graph of a piecewise function?

---

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

Q: What is the difference between a piecewise function and a continuous function?

---

A: A continuous function is a function that can be drawn without lifting the pencil from the paper, whereas a piecewise function is a function that is defined by multiple sub-functions, each of which applies to a specific interval of the domain.

Q: Can I have a piecewise function that is not continuous?

---

A: Yes, you can have a piecewise function that is not continuous. In fact, many piecewise functions are not continuous, especially if they have different sub-functions that apply to different intervals of the domain.

Q: How do I find the domain of a piecewise function?

---

A: To find the domain of a piecewise function, you need to find the intervals where each sub-function is defined and then combine them to form the overall domain of the function.

Q: Can I have a piecewise function with a variable in the denominator?

---

A: Yes, you can have a piecewise function with a variable in the denominator. However, you need to be careful when evaluating the function, as the denominator cannot be zero.

Q: How do I evaluate a piecewise function at a specific value?

---

A: To evaluate a piecewise function at a specific value, you need to check which sub-function is defined for that value and then use that sub-function to evaluate the function.

Q: Can I have a piecewise function with a trigonometric function?

---

A: Yes, you can have a piecewise function with a trigonometric function. In fact, many piecewise functions involve trigonometric functions, especially in the context of modeling real-world phenomena.

Q: How do I find the derivative of a piecewise function?

---

A: To find the derivative of a piecewise function, you need to find the derivative of each sub-function separately and then combine them to form the overall derivative of the function.

Q: Can I have a piecewise function with a logarithmic function?

---

A: Yes, you can have a piecewise function with a logarithmic function. In fact, many piecewise functions involve logarithmic functions, especially in the context of modeling real-world phenomena.

Q: How do I find the integral of a piecewise function?

---

A: To find the integral of a piecewise function, you need to find the integral of each sub-function separately and then combine them to form the overall integral of the function.

Conclusion

---

Piecewise functions are a powerful tool for modeling real-world phenomena and solving mathematical problems. By understanding how to work with piecewise functions, you can develop the skills you need to tackle a wide range of mathematical challenges.