Complete The Definition Of The Function $h(x)$ So That It Is Continuous Over Its Domain.$ h(x) = \left\{ \begin{array}{ll} x^3, & X \ \textless \ 0 \\ 2, & X = 0 \\ \sqrt{x}, & 0 \ \textless \ X \ \textless \ 4 \\ h, & X = 4

Introduction

In mathematics, a function is considered continuous if its graph can be drawn without lifting the pencil from the paper. This means that the function has no gaps or jumps in its graph, and it can be evaluated at any point in its domain. In this article, we will explore the concept of continuity and how to complete the definition of the function h(x) so that it is continuous over its domain.

What is Continuity?

Continuity is a fundamental concept in calculus that deals with the behavior of functions at a point or in a region. A function f(x) is said to be continuous at a point x = a if the following conditions are met:

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

In other words, a function is continuous at a point if it can be evaluated at that point and if the value of the function at that point is equal to the limit of the function as x approaches that point.

The Function h(x)

The function h(x) is defined as:

$$h(x) = \left\{ \begin{array}{ll} x^3, & x \ \textless \ 0 \\ 2, & x = 0 \\ \sqrt{x}, & 0 \ \textless \ x \ \textless \ 4 \\ h, & x = 4 \end{array} \right.$$

This function has four different definitions, each corresponding to a different interval in the domain of the function. The first definition is for x < 0, where the function is equal to x^3. The second definition is for x = 0, where the function is equal to 2. The third definition is for 0 < x < 4, where the function is equal to the square root of x. The fourth definition is for x = 4, where the function is equal to itself.

Completing the Definition of h(x)

To complete the definition of h(x) so that it is continuous over its domain, we need to determine the value of the function at x = 4. Since the function is equal to itself at x = 4, we can write:

$$h(4) = h$$

However, this is not a complete definition of the function, as it does not specify the value of the function at x = 4. To complete the definition, we need to determine the value of the function at x = 4.

Determining the Value of h(4)

To determine the value of h(4), we need to examine the behavior of the function as x approaches 4 from the left and the right. As x approaches 4 from the left, the function is equal to the square root of x, which approaches 2. As x approaches 4 from the right, the function is equal to itself, which is also equal to 2.

Since the function approaches the same value from both the left and the right, we can conclude that the function is continuous at x = 4. Therefore, the value of h(4) is equal to 2.

Conclusion

In conclusion, the function h(x) is continuous over its domain if we define it as:

$$h(x) = \left\{ \begin{array}{ll} x^3, & x \ \textless \ 0 \\ 2, & x = 0 \\ \sqrt{x}, & 0 \ \textless \ x \ \textless \ 4 \\ 2, & x = 4 \end{array} \right.$$

This definition completes the function h(x) so that it is continuous over its domain.

Example Use Cases

The function h(x) can be used in a variety of applications, such as:

• Modeling the behavior of a physical system over time
• Analyzing the behavior of a complex system
• Making predictions about the behavior of a system

Step-by-Step Solution

To complete the definition of the function h(x) so that it is continuous over its domain, follow these steps:

1. Examine the behavior of the function as x approaches 4 from the left and the right.
2. Determine the value of the function at x = 4.
3. Define the function at x = 4 using the value determined in step 2.

Mathematical Background

The concept of continuity is a fundamental idea in calculus that deals with the behavior of functions at a point or in a region. A function f(x) is said to be continuous at a point x = a if the following conditions are met:

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

Key Takeaways

• A function is continuous at a point if it can be evaluated at that point and if the value of the function at that point is equal to the limit of the function as x approaches that point.
• The function h(x) is continuous over its domain if we define it as:

$$h(x) = \left\{ \begin{array}{ll} x^3, & x \ \textless \ 0 \\ 2, & x = 0 \\ \sqrt{x}, & 0 \ \textless \ x \ \textless \ 4 \\ 2, & x = 4 \end{array} \right.$$

• To complete the definition of the function h(x) so that it is continuous over its domain, follow the steps outlined in the step-by-step solution.
  Q&A: Completing the Definition of the Function h(x) for Continuity ====================================================================

Q: What is the main goal of completing the definition of the function h(x)?

A: The main goal of completing the definition of the function h(x) is to make it continuous over its domain. This means that the function should be able to be evaluated at any point in its domain without any gaps or jumps in its graph.

Q: What are the conditions for a function to be continuous at a point?

A: A function f(x) is said to be continuous at a point x = a if the following conditions are met:

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

Q: How do we determine the value of the function at x = 4?

A: To determine the value of the function at x = 4, we need to examine the behavior of the function as x approaches 4 from the left and the right. As x approaches 4 from the left, the function is equal to the square root of x, which approaches 2. As x approaches 4 from the right, the function is equal to itself, which is also equal to 2.

Q: Why is it important to complete the definition of the function h(x)?

A: Completing the definition of the function h(x) is important because it ensures that the function is continuous over its domain. This is crucial in many applications, such as modeling the behavior of a physical system over time, analyzing the behavior of a complex system, and making predictions about the behavior of a system.

Q: What are some real-world applications of the function h(x)?

A: The function h(x) can be used in a variety of applications, such as:

• Modeling the behavior of a physical system over time
• Analyzing the behavior of a complex system
• Making predictions about the behavior of a system

Q: How do we complete the definition of the function h(x) so that it is continuous over its domain?

A: To complete the definition of the function h(x) so that it is continuous over its domain, follow these steps:

1. Examine the behavior of the function as x approaches 4 from the left and the right.
2. Determine the value of the function at x = 4.
3. Define the function at x = 4 using the value determined in step 2.

Q: What is the final definition of the function h(x) after completing it?

A: The final definition of the function h(x) after completing it is:

$$h(x) = \left\{ \begin{array}{ll} x^3, & x \ \textless \ 0 \\ 2, & x = 0 \\ \sqrt{x}, & 0 \ \textless \ x \ \textless \ 4 \\ 2, & x = 4 \end{array} \right.$$

Q: What are some key takeaways from this article?

A: Some key takeaways from this article are:

• A function is continuous at a point if it can be evaluated at that point and if the value of the function at that point is equal to the limit of the function as x approaches that point.
• The function h(x) is continuous over its domain if we define it as:

$$h(x) = \left\{ \begin{array}{ll} x^3, & x \ \textless \ 0 \\ 2, & x = 0 \\ \sqrt{x}, & 0 \ \textless \ x \ \textless \ 4 \\ 2, & x = 4 \end{array} \right.$$

• To complete the definition of the function h(x) so that it is continuous over its domain, follow the steps outlined in the step-by-step solution.