Evaluating The Piecewise Function H(x) At X=0

Jul 15, 2025 by Admin 46 views

Evaluating the Piecewise Function h(x) at Specific Points

In mathematics, a piecewise function is a function defined by multiple sub-functions, where each sub-function applies to a certain interval of the main function's domain. Understanding how to evaluate these functions at specific points is a fundamental skill. In this article, we will explore the evaluation of a piecewise function, denoted as h(x), at particular values of x. Specifically, we will determine the values of h(x) when x equals 0 and another unspecified value.

Understanding Piecewise Functions

Before diving into the specific problem, let's clarify the concept of piecewise functions. A piecewise function, as the name suggests, is defined in "pieces." Each piece is a different function that applies over a specific interval of the input variable, x. The intervals are usually defined using inequalities.

To evaluate a piecewise function at a particular value of x, you must first identify which interval x belongs to. Then, you apply the corresponding sub-function defined for that interval. This careful selection of the correct sub-function is crucial for obtaining the correct result. A deep understanding of the definition of piecewise functions is very important for solving related problems. Furthermore, mastering function evaluation techniques is an essential skill in mathematics, especially when dealing with piecewise functions. These functions, defined by different formulas over different intervals, require careful attention to detail when calculating their values at specific points.

Problem Statement

We are given the piecewise function h(x) defined as follows:

h(x) = 
\begin{cases}
3x - 4, & x < 0 \\
2x^2 - 3x + 10, & 0 \leq x < 4 \\
2^x, & x \geq 4
\end{cases}

This function has three sub-functions, each applicable over a different interval of x:

For x less than 0, h(x) is defined as 3x - 4.
For x between 0 (inclusive) and 4 (exclusive), h(x) is defined as 2x² - 3x + 10.
For x greater than or equal to 4, h(x) is defined as 2^x.

Our task is to find the values of h(x) when x = 0 and when x is another unspecified value. Let's tackle the case when x = 0 first.

Evaluating h(0)

To find h(0), we need to determine which interval x = 0 falls into. Looking at the definition of h(x), we see that the interval 0 ≤ x < 4 includes x = 0. Therefore, we use the corresponding sub-function:

h(x) = 2x² - 3x + 10

Substituting x = 0 into this sub-function, we get:

h(0) = 2(0)² - 3(0) + 10 = 0 - 0 + 10 = 10

Thus, h(0) = 10. This demonstrates the importance of selecting the correct sub-function based on the interval in which the input value lies. The process of evaluating piecewise functions at specific points involves identifying the appropriate interval and applying the corresponding sub-function. This careful approach ensures accurate results and a solid understanding of the function's behavior. By mastering these techniques, students can confidently tackle more complex mathematical problems involving piecewise functions. The ability to evaluate piecewise functions is a critical skill in various mathematical contexts, including calculus, analysis, and discrete mathematics.

Evaluating h(x) for another value

Now, let's consider the second part of the problem, where we need to evaluate h(x) for another unspecified value of x. Since the value of x is not provided, we cannot compute a numerical answer. However, we can discuss the process of evaluating h(x) for any given value of x. This involves a similar approach to what we used for h(0), but with a focus on the general methodology. The key is to first determine the interval in which the given value of x lies. Once we have identified the correct interval, we can apply the corresponding sub-function to calculate the value of h(x). For instance, if we were given a value of x less than 0, we would use the sub-function h(x) = 3x - 4. If the value of x were between 0 and 4, we would use h(x) = 2x^2 - 3x + 10. And if the value of x were greater than or equal to 4, we would use h(x) = 2^x. This step-by-step process ensures that we are always using the correct formula for the given input value. The process of interval identification is fundamental to the accurate evaluation of piecewise functions. This involves carefully examining the conditions that define each sub-function and determining which condition the input value satisfies. By systematically applying this approach, we can confidently evaluate h(x) for any given value of x, demonstrating a comprehensive understanding of piecewise function behavior. Understanding the domain and range of piecewise functions is also crucial for a complete understanding. Each sub-function is defined over a specific interval, and the overall domain of the piecewise function is the union of these intervals. Similarly, the range is determined by the values that each sub-function can produce within its interval. These concepts provide a broader context for evaluating piecewise functions and interpreting their results.

Scenario 1: Assume x is less than 0

If we assume x is less than 0 (i.e., x < 0), we would use the first sub-function:

h(x) = 3x - 4

In this case, the value of h(x) would depend on the specific value of x we choose. For example, if x = -2, then:

h(-2) = 3(-2) - 4 = -6 - 4 = -10

Scenario 2: Assume x is between 0 and 4

If we assume x is between 0 and 4 (i.e., 0 ≤ x < 4), we would use the second sub-function:

h(x) = 2x² - 3x + 10

Again, the value of h(x) would depend on the specific value of x. For example, if x = 2, then:

h(2) = 2(2)² - 3(2) + 10 = 8 - 6 + 10 = 12

Scenario 3: Assume x is greater than or equal to 4

Finally, if we assume x is greater than or equal to 4 (i.e., x ≥ 4), we would use the third sub-function:

h(x) = 2^x

In this case, the value of h(x) would grow exponentially as x increases. For example, if x = 4, then:

h(4) = 2⁴ = 16

Conclusion

In conclusion, we have successfully evaluated the piecewise function h(x) at x = 0, finding that h(0) = 10. We have also discussed the general procedure for evaluating h(x) for any given value of x, emphasizing the importance of identifying the correct interval and applying the corresponding sub-function. This methodical approach ensures accurate results and a thorough understanding of piecewise function behavior. By mastering the techniques outlined in this article, you can confidently tackle problems involving piecewise functions and further enhance your mathematical skills. The ability to evaluate functions at specific points is a fundamental concept in mathematics and is essential for understanding the behavior of various mathematical models. Piecewise functions, in particular, provide a versatile tool for representing situations where different rules apply under different conditions, making their evaluation a crucial skill in many areas of mathematics and its applications.