Let $g$ Be The Piecewise Defined Function Shown:${ G(x)=\begin{cases} x + 4, & -5 \leq X \leq -1 \ 2 - X, & -1 \ \textless \ X \leq 5 \end{cases} }$Evaluate $g$ At Different Values In Its Domain:$[

Introduction

Piecewise functions are a type of mathematical function that is defined by multiple sub-functions, each applied to a specific interval of the domain. These functions are commonly used in mathematics, physics, and engineering to model real-world phenomena. In this article, we will explore the concept of piecewise functions, with a focus on evaluating the function $g$ at different values in its domain.

The Piecewise Function $g$

The piecewise function $g$ is defined as:

{ g(x)=\begin{cases} x + 4, & -5 \leq x \leq -1 \\ 2 - x, & -1 \ \textless \ x \leq 5 \end{cases} \}

This function has two sub-functions: $f_1(x) = x + 4$ for $-5 \leq x \leq -1$, and $f_2(x) = 2 - x$ for $-1 \ \textless \ x \leq 5$.

Evaluating $g$ at Different Values in its Domain

To evaluate the function $g$ at different values in its domain, we need to determine which sub-function to use for each value of $x$. We will start by evaluating $g$ at some specific values in its domain.

Evaluating $g$ at $x = -3$

For $x = -3$, we have $-5 \leq -3 \leq -1$, so we use the sub-function $f_1(x) = x + 4$. Therefore, $g(-3) = f_1(-3) = -3 + 4 = 1$.

Evaluating $g$ at $x = 0$

For $x = 0$, we have $-1 \ \textless \ 0 \leq 5$, so we use the sub-function $f_2(x) = 2 - x$. Therefore, $g(0) = f_2(0) = 2 - 0 = 2$.

Evaluating $g$ at $x = 3$

For $x = 3$, we have $-1 \ \textless \ 3 \leq 5$, so we use the sub-function $f_2(x) = 2 - x$. Therefore, $g(3) = f_2(3) = 2 - 3 = -1$.

Evaluating $g$ at $x = -5$

For $x = -5$, we have $-5 \leq -5 \leq -1$, so we use the sub-function $f_1(x) = x + 4$. Therefore, $g(-5) = f_1(-5) = -5 + 4 = -1$.

Evaluating $g$ at $x = 5$

For $x = 5$, we have $-1 \ \textless \ 5 \leq 5$, so we use the sub-function $f_2(x) = 2 - x$. Therefore, $g(5) = f_2(5) = 2 - 5 = -3$.

Conclusion

In this article, we have evaluated the piecewise function $g$ at different values in its domain. We have shown that the function $g$ is defined by two sub-functions, $f_1(x) = x + 4$ and $f_2(x) = 2 - x$, each applied to a specific interval of the domain. We have also demonstrated how to evaluate the function $g$ at specific values in its domain by determining which sub-function to use for each value of $x$. This article provides a comprehensive guide to evaluating piecewise functions, with a focus on the function $g$.

Example Problems

Problem 1

Evaluate the function $g$ at $x = -2$.

Solution

For $x = -2$, we have $-5 \leq -2 \leq -1$, so we use the sub-function $f_1(x) = x + 4$. Therefore, $g(-2) = f_1(-2) = -2 + 4 = 2$.

Problem 2

Evaluate the function $g$ at $x = 4$.

Solution

For $x = 4$, we have $-1 \ \textless \ 4 \leq 5$, so we use the sub-function $f_2(x) = 2 - x$. Therefore, $g(4) = f_2(4) = 2 - 4 = -2$.

Applications of Piecewise Functions

Piecewise functions have numerous applications in mathematics, physics, and engineering. Some examples include:

• Modeling real-world phenomena: Piecewise functions can be used to model real-world phenomena, such as the motion of an object under the influence of gravity.
• Optimization problems: Piecewise functions can be used to solve optimization problems, such as finding the maximum or minimum value of a function.
• Signal processing: Piecewise functions can be used in signal processing to model and analyze signals.

Conclusion

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Q&A: Evaluating Piecewise Functions

Q: What is a piecewise function?

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

Q: How do I evaluate a piecewise function?

A: To evaluate a piecewise function, you need to determine which sub-function to use for each value of x. You do this by checking which interval the value of x falls into.

Q: What if the value of x falls into multiple intervals?

A: If the value of x falls into multiple intervals, you need to use the sub-function that is defined for the interval that x falls into. If x falls into multiple intervals with different sub-functions, you need to use the sub-function that is defined for the interval that x falls into first.

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

A: To determine which sub-function to use, you need to check the intervals that the value of x falls into. You can do this by looking at the piecewise function definition and checking which interval the value of x falls into.

Q: What if I'm not sure which sub-function to use?

A: If you're not sure which sub-function to use, you can try plugging in the value of x into each sub-function and see which one gives you a valid result.

Q: Can I use piecewise functions to model real-world phenomena?

A: Yes, piecewise functions can be used to model real-world phenomena. They can be used to model the motion of an object under the influence of gravity, the growth of a population, and many other real-world phenomena.

Q: Can I use piecewise functions to solve optimization problems?

A: Yes, piecewise functions can be used to solve optimization problems. They can be used to find the maximum or minimum value of a function.

Q: Can I use piecewise functions in signal processing?

A: Yes, piecewise functions can be used in signal processing. They can be used to model and analyze signals.

Q: How do I graph a piecewise function?

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

Q: Can I use piecewise functions to model periodic phenomena?

A: Yes, piecewise functions can be used to model periodic phenomena. They can be used to model the motion of a pendulum, the growth of a population, and many other periodic phenomena.

Q: Can I use piecewise functions to model non-periodic phenomena?

A: Yes, piecewise functions can be used to model non-periodic phenomena. They can be used to model the growth of a population, the motion of an object under the influence of gravity, and many other non-periodic phenomena.

Conclusion

In conclusion, piecewise functions are a powerful tool in mathematics, physics, and engineering. They can be used to model real-world phenomena, solve optimization problems, and analyze signals. This article has provided a comprehensive guide to evaluating piecewise functions, with a focus on the function g. We have demonstrated how to evaluate the function g at different values in its domain and have shown the importance of piecewise functions in various applications.

Example Problems

Problem 1

Evaluate the function g at x = -2.

Solution

For x = -2, we have -5 ≤ -2 ≤ -1, so we use the sub-function f1(x) = x + 4. Therefore, g(-2) = f1(-2) = -2 + 4 = 2.

Problem 2

Evaluate the function g at x = 4.

Solution

For x = 4, we have -1 < 4 ≤ 5, so we use the sub-function f2(x) = 2 - x. Therefore, g(4) = f2(4) = 2 - 4 = -2.

Applications of Piecewise Functions

Piecewise functions have numerous applications in mathematics, physics, and engineering. Some examples include:

• Modeling real-world phenomena: Piecewise functions can be used to model real-world phenomena, such as the motion of an object under the influence of gravity.
• Optimization problems: Piecewise functions can be used to solve optimization problems, such as finding the maximum or minimum value of a function.
• Signal processing: Piecewise functions can be used in signal processing to model and analyze signals.

Conclusion

In conclusion, piecewise functions are a powerful tool in mathematics, physics, and engineering. They can be used to model real-world phenomena, solve optimization problems, and analyze signals. This article has provided a comprehensive guide to evaluating piecewise functions, with a focus on the function g. We have demonstrated how to evaluate the function g at different values in its domain and have shown the importance of piecewise functions in various applications.