Understanding the Intervals

Oct 13, 2025 by windailys.com 93 views

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 often used to model real-world scenarios where the behavior of a system changes at certain points. In this article, we will explore the piecewise-defined function $f(x)$ and determine the range of values it can take.

The Piecewise-Defined Function

The given piecewise-defined function is:

f(x) = \begin{cases} 2x + 2, & x < -3 \\ x, & x = -3 \\ -x - 2, & x > -3 \end{cases}

This function has three sub-functions, each defined on a specific interval of the domain. The first sub-function, $2x + 2$ , is defined for $x < -3$ . The second sub-function, $x$ , is defined at $x = -3$ . The third sub-function, $-x - 2$ , is defined for $x > -3$ .

Understanding the Intervals

To evaluate the range of values of the piecewise-defined function, we need to understand the intervals on which each sub-function is defined. The first sub-function, $2x + 2$ , is defined for $x < -3$ . This means that for any value of $x$ less than $-3$ , the function will take on the value of $2x + 2$ . The second sub-function, $x$ , is defined at $x = -3$ . This means that at $x = -3$ , the function will take on the value of $x$ , which is $-3$ . The third sub-function, $-x - 2$ , is defined for $x > -3$ . This means that for any value of $x$ greater than $-3$ , the function will take on the value of $-x - 2$ .

Evaluating the Range of Values

To determine the range of values of the piecewise-defined function, we need to evaluate the function at different points in its domain. Let's start by evaluating the function at $x = -4$ .

f(-4) = 2(-4) + 2 = -6

Since $-4 < -3$ , the function takes on the value of $2x + 2$ at $x = -4$ . Therefore, the value of the function at $x = -4$ is $-6$ .

Next, let's evaluate the function at $x = -3$ .

f(-3) = -(-3) - 2 = -1

Since $x = -3$ , the function takes on the value of $x$ at $x = -3$ . Therefore, the value of the function at $x = -3$ is $-1$ .

Finally, let's evaluate the function at $x = -2$ .

f(-2) = -(-2) - 2 = 0

Since $-2 > -3$ , the function takes on the value of $-x - 2$ at $x = -2$ . Therefore, the value of the function at $x = -2$ is $0$ .

Conclusion

In conclusion, the piecewise-defined function $f(x)$ has a range of values that includes $-6$ , $-1$ , and $0$ . These values are obtained by evaluating the function at different points in its domain. The function takes on the value of $2x + 2$ for $x < -3$ , the value of $x$ at $x = -3$ , and the value of $-x - 2$ for $x > -3$ .

Range of Values

The range of values of the piecewise-defined function $f(x)$ is:

\{-6, -1, 0\}

This means that the function can take on the values of $-6$ , $-1$ , and $0$ .

Options

The options for the range of values of the piecewise-defined function $f(x)$ are:

$y = -6$
$y = -1$
$y = 0$

The correct answer is:

$y = -6$

This is because the function takes on the value of $2x + 2$ for $x < -3$ , and $2(-4) + 2 = -6$ .

Discussion

The piecewise-defined function $f(x)$ is a useful tool for modeling real-world scenarios where the behavior of a system changes at certain points. The function has three sub-functions, each defined on a specific interval of the domain. The first sub-function, $2x + 2$ , is defined for $x < -3$ . The second sub-function, $x$ , is defined at $x = -3$ . The third sub-function, $-x - 2$ , is defined for $x > -3$ .

The range of values of the piecewise-defined function $f(x)$ is $\{-6, -1, 0\}$ . This means that the function can take on the values of $-6$ , $-1$ , and $0$ . The correct answer is $y = -6$ , which is obtained by evaluating the function at $x = -4$ .

Final Answer

The final answer is:

$y = -6$
Q&A: Piecewise-Defined Function =====================================

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 determine the range of values of a piecewise-defined function?

A: To determine the range of values of a piecewise-defined function, you need to evaluate the function at different points in its domain. You should consider the intervals on which each sub-function is defined and evaluate the function at a point in each interval.

Q: What is the difference between a piecewise-defined function and a continuous 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. A continuous function, on the other hand, is a function that can be drawn without lifting the pencil from the paper. In other words, a continuous function has no gaps or jumps in its graph.

Q: Can a piecewise-defined function be continuous?

A: Yes, a piecewise-defined function can be continuous. For example, the function $f(x) = \begin{cases} 2x + 2, & x < 0 \\ x, & x \geq 0 \end{cases}$ is a piecewise-defined function that is continuous.

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 the graphs. You should consider the intervals on which each sub-function is defined and graph the function on each interval.

Q: Can a piecewise-defined function have multiple intervals?

A: Yes, a piecewise-defined function can have multiple intervals. For example, the function $f(x) = \begin{cases} 2x + 2, & x < -3 \\ x, & -3 \leq x < 0 \\ -x - 2, & x \geq 0 \end{cases}$ is a piecewise-defined function with three intervals.

Q: How do I evaluate a piecewise-defined function at a point?

A: To evaluate a piecewise-defined function at a point, you need to determine which sub-function is defined at that point and evaluate the function using that sub-function.

Q: Can a piecewise-defined function be used to model real-world scenarios?

A: Yes, a piecewise-defined function can be used to model real-world scenarios where the behavior of a system changes at certain points. For example, the function $f(x) = \begin{cases} 2x + 2, & x < 0 \\ x, & x \geq 0 \end{cases}$ can be used to model the behavior of a system that changes at $x = 0$ .

Q: How do I determine the domain of a piecewise-defined function?

A: To determine the domain of a piecewise-defined function, you need to consider the intervals on which each sub-function is defined and determine the intersection of those intervals.

Q: Can a piecewise-defined function have a domain that is a single interval?

A: Yes, a piecewise-defined function can have a domain that is a single interval. For example, the function $f(x) = \begin{cases} 2x + 2, & x < 0 \end{cases}$ is a piecewise-defined function with a domain that is a single interval.

Q: How do I determine the range of a piecewise-defined function?

A: To determine the range of a piecewise-defined function, you need to evaluate the function at different points in its domain and determine the set of all possible output values.

Q: Can a piecewise-defined function have a range that is a single value?

A: Yes, a piecewise-defined function can have a range that is a single value. For example, the function $f(x) = \begin{cases} 2x + 2, & x < 0 \\ x, & x \geq 0 \end{cases}$ has a range that is a single value, $0$ .

Q: How do I determine the maximum and minimum values of a piecewise-defined function?

A: To determine the maximum and minimum values of a piecewise-defined function, you need to evaluate the function at different points in its domain and determine the largest and smallest output values.

Q: Can a piecewise-defined function have multiple maximum and minimum values?

A: Yes, a piecewise-defined function can have multiple maximum and minimum values. For example, the function $f(x) = \begin{cases} 2x + 2, & x < 0 \\ x, & x \geq 0 \end{cases}$ has multiple maximum and minimum values.

Q: How do I determine the symmetry of a piecewise-defined function?

A: To determine the symmetry of a piecewise-defined function, you need to evaluate the function at different points in its domain and determine if the function is symmetric about a particular point or line.

Q: Can a piecewise-defined function have symmetry?

A: Yes, a piecewise-defined function can have symmetry. For example, the function $f(x) = \begin{cases} 2x + 2, & x < 0 \\ x, & x \geq 0 \end{cases}$ has symmetry about the point $(0, 0)$ .

Q: How do I determine the asymptotes of a piecewise-defined function?

A: To determine the asymptotes of a piecewise-defined function, you need to evaluate the function at different points in its domain and determine if the function approaches a particular value or line as the input values approach a certain point or infinity.

Q: Can a piecewise-defined function have asymptotes?

A: Yes, a piecewise-defined function can have asymptotes. For example, the function $f(x) = \begin{cases} 2x + 2, & x < 0 \\ x, & x \geq 0 \end{cases}$ has asymptotes at $x = 0$ and $y = 0$ .

Q: How do I determine the derivative of a piecewise-defined function?

A: To determine the derivative of a piecewise-defined function, you need to evaluate the function at different points in its domain and determine the rate of change of the function at each point.

Q: Can a piecewise-defined function have a derivative?

A: Yes, a piecewise-defined function can have a derivative. For example, the function $f(x) = \begin{cases} 2x + 2, & x < 0 \\ x, & x \geq 0 \end{cases}$ has a derivative at each point in its domain.

Q: How do I determine the integral of a piecewise-defined function?

A: To determine the integral of a piecewise-defined function, you need to evaluate the function at different points in its domain and determine the area under the curve of the function.

Q: Can a piecewise-defined function have an integral?

A: Yes, a piecewise-defined function can have an integral. For example, the function $f(x) = \begin{cases} 2x + 2, & x < 0 \\ x, & x \geq 0 \end{cases}$ has an integral that can be evaluated using the fundamental theorem of calculus.