Which Value Is In The Domain Of $f(x$\]?$f(x) = \begin{cases} 2x + 5, & -6 \ \textless \ X \leq 0 \\ -2x + 3, & 0 \ \textless \ X \leq 4 \end{cases}$A. -7 B. -6 C. 4 D. 5

A piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. In this article, we will explore the domain of a piecewise function and determine which value is in the domain of the given function.

Understanding Piecewise Functions

A piecewise 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 using the following notation:

$$f(x) = \begin{cases} f_1(x), & x \in I_1 \\ f_2(x), & x \in I_2 \\ \vdots \\ f_n(x), & x \in I_n \end{cases}$$

where $f_1(x), f_2(x), \ldots, f_n(x)$ are the sub-functions, and $I_1, I_2, \ldots, I_n$ are the intervals of the domain.

The Given Function

The given function is defined as:

$$f(x) = \begin{cases} 2x + 5, & -6 \lt x \leq 0 \\ -2x + 3, & 0 \lt x \leq 4 \end{cases}$$

This function is a piecewise function with two sub-functions: $f_1(x) = 2x + 5$ and $f_2(x) = -2x + 3$. The function is defined on two intervals: $-6 \lt x \leq 0$ and $0 \lt x \leq 4$.

Determining the Domain

To determine the domain of the function, we need to consider the intervals on which the function is defined. In this case, the function is defined on two intervals: $-6 \lt x \leq 0$ and $0 \lt x \leq 4$.

The domain of the function is the set of all values of $x$ for which the function is defined. In this case, the domain is the union of the two intervals:

$$\text{Domain} = (-6, 0] \cup (0, 4]$$

Which Value is in the Domain?

Now that we have determined the domain of the function, we can determine which value is in the domain. The options are:

A. -7 B. -6 C. 4 D. 5

To determine which value is in the domain, we need to check if each value is in the domain. We can do this by plugging each value into the function and checking if it is defined.

• For option A, -7 is not in the domain because it is less than -6.
• For option B, -6 is not in the domain because it is not in the interval $-6 \lt x \leq 0$.
• For option C, 4 is in the domain because it is in the interval $0 \lt x \leq 4$.
• For option D, 5 is not in the domain because it is greater than 4.

Therefore, the value that is in the domain of the function is C. 4.

Conclusion

In the previous article, we explored the domain of a piecewise function and determined which value is in the domain of the given function. In this article, we will answer some frequently asked questions (FAQs) about piecewise functions.

Q: What is a piecewise function?

A piecewise 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 domain of a piecewise function?

To determine the domain of a piecewise function, you need to consider the intervals on which the function is defined. The domain of the function is the set of all values of $x$ for which the function is defined.

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

A piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. A regular function, on the other hand, is a function that is defined by a single formula for all values of $x$.

Q: Can a piecewise function have more than two sub-functions?

Yes, a piecewise function can have more than two sub-functions. In fact, a piecewise function can have any number of sub-functions, each applied to a specific interval of the domain.

Q: How do I graph a piecewise function?

To graph a piecewise function, you need to graph each sub-function separately, using the corresponding interval of the domain. You can then combine the graphs to form the graph of the piecewise function.

Q: Can a piecewise function be continuous?

Yes, a piecewise function can be continuous. In fact, a piecewise function can be continuous if the sub-functions are continuous and the intervals of the domain are connected.

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

To find the derivative of a piecewise function, you need to find the derivative of each sub-function separately, using the corresponding interval of the domain. You can then combine the derivatives to form the derivative of the piecewise function.

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

Yes, a piecewise function can be used to model real-world phenomena. In fact, piecewise functions are often used to model situations where the behavior of a system changes abruptly at certain points.

Q: What are some common applications of piecewise functions?

Some common applications of piecewise functions include:

• Modeling population growth and decline
• Modeling the behavior of electrical circuits
• Modeling the behavior of mechanical systems
• Modeling the behavior of financial systems

Conclusion

In this article, we answered some frequently asked questions (FAQs) about piecewise functions. We learned that piecewise functions are functions that are defined by multiple sub-functions, each applied to a specific interval of the domain. We also learned how to determine the domain of a piecewise function, how to graph a piecewise function, and how to find the derivative of a piecewise function.