Complete The Definition Of H ( X H(x H ( X ] So That It Is Continuous Over Its Domain.$ h(x)=\begin{cases} x^3, & X \ \textless \ 0 \ a, & X = 0 \ \sqrt{x}, & 0 \ \textless \ X \ \textless \ 4 \ b, & X = 4 \ 4-\frac{1}{2} X, & X \

Introduction

In mathematics, a piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval or domain. These sub-functions are often referred to as "pieces" of the overall function. When working with piecewise functions, it is essential to ensure that the function is continuous over its domain. In this article, we will explore how to complete the definition of a piecewise function, specifically the function $h(x)$, so that it is continuous over its domain.

The Function $h(x)$

The function $h(x)$ is defined as follows:

$$h(x)=\begin{cases} x^3, & x \ \textless \ 0 \\ a, & x = 0 \\ \sqrt{x}, & 0 \ \textless \ x \ \textless \ 4 \\ b, & x = 4 \\ 4-\frac{1}{2} x, & x \ \end{cases}$$

As we can see, the function $h(x)$ is defined by five different sub-functions, each applied to a specific interval or domain. The intervals are:

• $x < 0$
• $x = 0$
• $0 < x < 4$
• $x = 4$
• $x > 4$

Continuity of a Function

A function is said to be continuous at a point if the following conditions are met:

1. The function is defined at the point.
2. The limit of the function as $x$ approaches the point exists.
3. The limit of the function as $x$ approaches the point is equal to the value of the function at the point.

In other words, a function is continuous at a point if it can be drawn without lifting the pencil from the paper.

Completing the Definition of $h(x)$

To complete the definition of $h(x)$, we need to ensure that the function is continuous over its domain. This means that we need to find the values of $a$ and $b$ such that the function is continuous at $x = 0$ and $x = 4$.

Continuity at $x = 0$

To ensure continuity at $x = 0$, we need to find the value of $a$ such that:

$$\lim_{x \to 0^-} h(x) = \lim_{x \to 0^+} h(x) = h(0)$$

Since $h(x) = x^3$ for $x < 0$, we have:

$$\lim_{x \to 0^-} h(x) = \lim_{x \to 0^-} x^3 = 0$$

Since $h(x) = \sqrt{x}$ for $0 < x < 4$, we have:

$$\lim_{x \to 0^+} h(x) = \lim_{x \to 0^+} \sqrt{x} = 0$$

Therefore, we need to find the value of $a$ such that:

$$a = 0$$

Continuity at $x = 4$

To ensure continuity at $x = 4$, we need to find the value of $b$ such that:

$$\lim_{x \to 4^-} h(x) = \lim_{x \to 4^+} h(x) = h(4)$$

Since $h(x) = \sqrt{x}$ for $0 < x < 4$, we have:

$$\lim_{x \to 4^-} h(x) = \lim_{x \to 4^-} \sqrt{x} = 2$$

Since $h(x) = 4 - \frac{1}{2}x$ for $x > 4$, we have:

$$\lim_{x \to 4^+} h(x) = \lim_{x \to 4^+} 4 - \frac{1}{2}x = 2$$

Therefore, we need to find the value of $b$ such that:

$$b = 2$$

The Completed Definition of $h(x)$

With the values of $a$ and $b$ determined, we can now complete the definition of $h(x)$:

$$h(x)=\begin{cases} x^3, & x \ \textless \ 0 \\ 0, & x = 0 \\ \sqrt{x}, & 0 \ \textless \ x \ \textless \ 4 \\ 2, & x = 4 \\ 4-\frac{1}{2} x, & x \ \end{cases}$$

Conclusion

In this article, we have explored how to complete the definition of a piecewise function, specifically the function $h(x)$, so that it is continuous over its domain. We have determined the values of $a$ and $b$ such that the function is continuous at $x = 0$ and $x = 4$. The completed definition of $h(x)$ is:

$$h(x)=\begin{cases} x^3, & x \ \textless \ 0 \\ 0, & x = 0 \\ \sqrt{x}, & 0 \ \textless \ x \ \textless \ 4 \\ 2, & x = 4 \\ 4-\frac{1}{2} x, & x \ \end{cases}$$

Introduction

In our previous article, we explored how to complete the definition of a piecewise function, specifically the function $h(x)$, so that it is continuous over its domain. We determined the values of $a$ and $b$ such that the function is continuous at $x = 0$ and $x = 4$. In this article, we will answer some frequently asked questions about completing the definition of a piecewise function for continuity.

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 or domain. These sub-functions are often referred to as "pieces" of the overall function.

Q: Why is it important to ensure continuity of a piecewise function?

Ensuring continuity of a piecewise function is important because it allows us to model real-world phenomena that exhibit piecewise behavior. If a function is not continuous, it may not accurately represent the behavior of the phenomenon being modeled.

Q: How do I determine the values of $a$ and $b$ for a piecewise function?

To determine the values of $a$ and $b$ for a piecewise function, you need to find the limits of the function as $x$ approaches the points where the function changes. You can use the following steps:

1. Identify the points where the function changes.
2. Find the limits of the function as $x$ approaches each of these points.
3. Set the limits equal to the values of the function at the points.
4. Solve for the values of $a$ and $b$.

Q: What if the limits of the function do not exist?

If the limits of the function do not exist, it may not be possible to determine the values of $a$ and $b$. In this case, you may need to re-examine the function and determine if it is possible to make it continuous.

Q: Can a piecewise function be continuous at a point where the function changes?

Yes, a piecewise function can be continuous at a point where the function changes. This is known as a "jump discontinuity." However, in this case, the function is not continuous in the classical sense, but rather it has a "jump" at the point where the function changes.

Q: How do I know if a piecewise function is continuous?

To determine if a piecewise function is continuous, you can use the following steps:

1. Check if the function is defined at each point in its domain.
2. Check if the limits of the function as $x$ approaches each point exist.
3. Check if the limits of the function as $x$ approaches each point are equal to the value of the function at the point.
4. If all of these conditions are met, the function is continuous.

Q: Can a piecewise function be continuous over its entire domain?

Yes, a piecewise function can be continuous over its entire domain. This is known as a "continuous piecewise function." However, it may require the use of more complex mathematical techniques to determine the values of $a$ and $b$.

Conclusion

In this article, we have answered some frequently asked questions about completing the definition of a piecewise function for continuity. We have discussed the importance of ensuring continuity of a piecewise function, how to determine the values of $a$ and $b$, and how to know if a piecewise function is continuous. We hope that this article has been helpful in understanding the concept of completing the definition of a piecewise function for continuity.