Complete The Definition Of The Function H(x) So That It Is Continuous Over Its Domain. The Function H(x) Is Defined Piecewise. Find The Values Of Constants A And B Such That H(x) Is Continuous At X = 0 And X = 4.

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In the fascinating world of calculus, continuity stands as a cornerstone concept. A function is said to be continuous if its graph can be drawn without lifting the pen, implying there are no abrupt jumps, breaks, or holes. This property is crucial for many mathematical operations and applications. In this article, we delve into the concept of continuity with a particular focus on piecewise functions. Our main objective is to determine the values of constants that will ensure a piecewise function is continuous across its entire domain. We will explore how to analyze the behavior of the function at the points where its definition changes, ensuring that the different pieces connect smoothly. This involves understanding the limits of the function as it approaches these critical points from both sides and ensuring that these limits match the function's value at the point. By mastering these techniques, we can confidently manipulate and define piecewise functions to meet the demands of various mathematical models and real-world applications. This article provides a detailed walkthrough of the process, including the necessary calculations and reasoning, to make this concept accessible and understandable. Whether you're a student learning calculus or a professional needing a refresher, this guide will equip you with the knowledge to tackle continuity problems effectively.

Let's consider the piecewise function defined as follows:

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

The challenge at hand is to find the values of the constants a and b such that the function h(x) is continuous over its entire domain. This means we need to ensure that the function's pieces connect seamlessly at the points where the definition changes, specifically at x = 0 and x = 4. To achieve this, we must delve into the concept of limits and how they relate to continuity. A function is continuous at a point if the limit of the function as x approaches that point exists, and this limit is equal to the function's value at that point. This requirement necessitates checking the left-hand limit, the right-hand limit, and the function's value at the point in question. For the function h(x), we will meticulously examine the behavior around x = 0 and x = 4, setting up equations based on the continuity conditions. Solving these equations will yield the specific values of a and b that ensure the function has no breaks or jumps, thereby making it continuous. This process illustrates a fundamental technique in calculus for handling piecewise functions and ensuring their smooth behavior across their domains. The following sections will walk through the steps in detail, providing clarity and a strong understanding of the underlying principles.

To ensure that the function h(x) is continuous, we need to satisfy the continuity conditions at the points where the function definition changes, which are x = 0 and x = 4. The fundamental principle of continuity at a point c states that a function f(x) is continuous at x = c if the following three conditions are met:

  1. f(c) is defined.
  2. The limit of f(x) as x approaches c exists, i.e., lim(x→c) f(x) exists.
  3. The limit of f(x) as x approaches c is equal to f(c), i.e., lim(x→c) f(x) = f(c).

For a piecewise function, we need to pay particular attention to the second condition, which requires the limit to exist. For a limit to exist at a point, the left-hand limit (the limit as x approaches c from the left, denoted as lim(x→c⁻) f(x)) must be equal to the right-hand limit (the limit as x approaches c from the right, denoted as lim(x→c⁺) f(x)). This ensures that the function approaches the same value from both sides of the point. At x = 0, we need to check that the left-hand limit of h(x) as x approaches 0 is equal to the right-hand limit of h(x) as x approaches 0, and that both are equal to h(0). Similarly, at x = 4, we need to check that the left-hand limit of h(x) as x approaches 4 is equal to the right-hand limit of h(x) as x approaches 4, and that both are equal to h(4). By applying these conditions systematically, we can determine the values of a and b that will make h(x) continuous. The next sections will detail the calculations for each point, illustrating how to apply these continuity principles in practice. These steps are crucial for understanding and manipulating piecewise functions effectively.

Continuity at x = 0

To ensure continuity at x = 0, we need to verify that the left-hand limit, the right-hand limit, and the function value at x = 0 are all equal. Let's break this down step by step.

1. Left-Hand Limit:

The left-hand limit is the limit of h(x) as x approaches 0 from the left (i.e., x < 0). In this region, h(x) = x³. Therefore, we need to calculate:

lim (x→0⁻) h(x) = lim (x→0⁻) x³

As x approaches 0 from the left, x³ also approaches 0. Thus,

lim (x→0⁻) x³ = 0

2. Right-Hand Limit:

The right-hand limit is the limit of h(x) as x approaches 0 from the right (i.e., 0 < x < 4). In this region, h(x) = √x. Therefore, we need to calculate:

lim (x→0⁺) h(x) = lim (x→0⁺) √x

As x approaches 0 from the right, √x also approaches 0. Thus,

lim (x→0⁺) √x = 0

3. Function Value at x = 0:

The function value at x = 0 is given by h(0) = a. For continuity, this value must be equal to both the left-hand limit and the right-hand limit.

4. Applying the Continuity Condition:

For h(x) to be continuous at x = 0, we need:

lim (x→0⁻) h(x) = lim (x→0⁺) h(x) = h(0)

Substituting the values we found:

0 = 0 = a

Therefore, we find that a = 0. This ensures that the function seamlessly transitions from x³ to √x at the point x = 0. By setting a to 0, we have satisfied the continuity condition at this critical point. The next section will focus on applying a similar process to ensure continuity at x = 4. Understanding this step-by-step approach is crucial for dealing with continuity in piecewise functions. The ability to calculate limits and apply the continuity condition is a fundamental skill in calculus, essential for analyzing the behavior of functions and their applications in various mathematical models.

Continuity at x = 4

Now, let's ensure the continuity of the function h(x) at x = 4. Following the same principles as before, we need to verify that the left-hand limit, the right-hand limit, and the function value at x = 4 are all equal. This involves analyzing the function's behavior as x approaches 4 from both the left and the right, and then comparing these limits with the defined value of the function at x = 4.

1. Left-Hand Limit:

The left-hand limit is the limit of h(x) as x approaches 4 from the left (i.e., 0 < x < 4). In this interval, h(x) = √x. So, we need to calculate:

lim (x→4⁻) h(x) = lim (x→4⁻) √x

As x approaches 4 from the left, √x approaches √4, which is 2. Thus,

lim (x→4⁻) √x = 2

2. Right-Hand Limit:

The right-hand limit is the limit of h(x) as x approaches 4 from the right (i.e., x > 4). In this region, h(x) = 4 - x/2*. Therefore, we need to calculate:

lim (x→4⁺) h(x) = lim (x→4⁺) (4 - x/2)

As x approaches 4 from the right, 4 - x/2 approaches 4 - 4/2 = 4 - 2 = 2. Thus,

lim (x→4⁺) (4 - x/2) = 2

3. Function Value at x = 4:

The function value at x = 4 is given by h(4) = b. For continuity, this value must be equal to both the left-hand limit and the right-hand limit.

4. Applying the Continuity Condition:

For h(x) to be continuous at x = 4, we need:

lim (x→4⁻) h(x) = lim (x→4⁺) h(x) = h(4)

Substituting the values we found:

2 = 2 = b

Therefore, we find that b = 2. This ensures that the function seamlessly transitions from √x to 4 - x/2 at the point x = 4. By setting b to 2, we have satisfied the continuity condition at this critical point as well. The meticulous examination of limits from both sides of x = 4 illustrates the importance of this technique in dealing with piecewise functions. Understanding and applying the continuity conditions at transition points is essential for ensuring the smooth behavior of functions, which is crucial in many mathematical and real-world applications. With both a and b determined, the function h(x) is now fully defined and continuous across its domain.

By applying the continuity conditions at x = 0 and x = 4, we have determined the values of the constants a and b that make the piecewise function h(x) continuous over its entire domain. At x = 0, we found that the left-hand limit, the right-hand limit, and the function value h(0) all needed to be equal. This led us to the equation a = 0. This ensures a smooth transition between the function segments x³ and √x at this critical point. Similarly, at x = 4, we analyzed the left-hand limit, the right-hand limit, and the function value h(4). This analysis resulted in the equation b = 2, ensuring a continuous connection between the function segments √x and 4 - x/2. Therefore, the final values that ensure the continuity of h(x) are:

  • a = 0
  • b = 2

With these values, the piecewise function h(x) is now defined as:

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

This comprehensive solution demonstrates the application of limits and continuity conditions to ensure that a piecewise function behaves predictably and smoothly across its domain. Understanding these principles is essential for various applications in calculus and real-world modeling, where the seamless transition between different functional behaviors is often required. The process of evaluating limits from both sides and ensuring they match the function value at the point is a cornerstone of continuity analysis. This detailed walkthrough provides a solid foundation for tackling similar problems and understanding the broader implications of continuity in mathematical functions.

In conclusion, ensuring the continuity of a piecewise function involves a rigorous examination of the function's behavior at the points where its definition changes. By applying the fundamental principles of limits and continuity conditions, we can determine the specific values of constants that guarantee a seamless transition between different function segments. This article has demonstrated a step-by-step approach to achieving this, focusing on the piecewise function h(x) and the determination of constants a and b. The key steps involve calculating the left-hand and right-hand limits at the critical points, setting these limits equal to each other and to the function's value at those points, and solving the resulting equations. For h(x), we found that setting a = 0 ensures continuity at x = 0, and setting b = 2 ensures continuity at x = 4. This process not only makes the function continuous but also provides a deeper understanding of the function's behavior across its domain. Understanding continuity is crucial in calculus and has far-reaching implications in various fields, including physics, engineering, and economics, where continuous models are often used to represent real-world phenomena. The ability to analyze and manipulate piecewise functions to ensure continuity is a valuable skill for anyone working with mathematical models. This article has provided a clear and thorough guide to this process, equipping readers with the knowledge and techniques necessary to tackle similar problems effectively. The meticulous approach outlined here emphasizes the importance of attention to detail and a strong grasp of the fundamental concepts of limits and continuity, which are essential for success in calculus and beyond.