Evaluating H(30) For H(t) = √(5-t) A Comprehensive Guide

In the realm of mathematics, functions serve as fundamental building blocks, mapping inputs to corresponding outputs. One such function is presented here: h(t) = √(5 - t). This function takes an input t, subtracts it from 5, and then calculates the square root of the result. Our task is to determine the value of this function when the input t is equal to 30. This involves substituting 30 for t in the function's expression and simplifying the resulting expression. This exploration will not only enhance your understanding of function evaluation but also solidify your grasp of basic arithmetic operations and the concept of square roots. Through a detailed step-by-step approach, we will unravel the process of evaluating h(30), ensuring clarity and comprehension at every stage. Join us as we delve into the intricacies of this mathematical problem, demystifying the process and empowering you to tackle similar challenges with confidence.

Before we dive into evaluating h(30), let's take a moment to fully understand the function itself. The function h(t) = √(5 - t) is defined as the square root of the difference between 5 and the input t. This means that for any given value of t, we first subtract it from 5, and then we find the square root of the result. Understanding this order of operations is crucial for accurate evaluation. The function's structure highlights the importance of the order of operations in mathematics. First, we perform the subtraction within the parentheses, and then we apply the square root operation. This careful sequencing ensures that we arrive at the correct output for a given input. Furthermore, the function introduces the concept of a square root, which is a fundamental operation in mathematics. Understanding square roots is essential for working with various mathematical concepts, including geometry, algebra, and calculus. This function serves as a practical example of how square roots are used in mathematical expressions, providing a concrete context for learning and applying this important concept. By grasping the components of this function, we lay a solid foundation for successfully evaluating it at t = 30.

Now, let's embark on the step-by-step evaluation of h(30). This process involves substituting 30 for t in the function's expression and then simplifying the result. Here's how it unfolds:

1. Substitution:

   • Replace t with 30 in the expression for h(t):
     • h(30) = √(5 - 30)

2. Subtraction:

   • Perform the subtraction inside the square root:
     • h(30) = √(-25)

3. Square Root of a Negative Number:

   • At this point, we encounter a crucial observation: we are attempting to find the square root of a negative number, -25.
   • In the realm of real numbers, the square root of a negative number is undefined. This is because no real number, when multiplied by itself, yields a negative result.
   • Therefore, √(-25) is not a real number.

4. Conclusion:

   • Since the square root of -25 is not a real number, we conclude that h(30) is undefined in the real number system.

This step-by-step approach highlights the importance of careful substitution and adherence to mathematical rules. By breaking down the evaluation into manageable steps, we avoid errors and gain a deeper understanding of the process. The encounter with the square root of a negative number serves as a valuable reminder of the limitations of the real number system and the existence of complex numbers, which extend the concept of numbers to include the square roots of negative numbers. This detailed evaluation not only provides the answer but also reinforces essential mathematical concepts.

The result that h(30) is undefined in the real number system holds significant implications. It underscores the crucial concept of the domain of a function. The domain of a function is the set of all possible input values for which the function produces a valid output. In the case of h(t) = √(5 - t), the domain is restricted to values of t that make the expression inside the square root non-negative. This is because the square root of a negative number is not a real number. Therefore, for h(t) to be defined, we must have:

5 - t ≥ 0

Solving this inequality for t, we get:

t ≤ 5

This inequality tells us that the function h(t) is only defined for values of t that are less than or equal to 5. Since 30 is greater than 5, it falls outside the domain of the function, and thus h(30) is undefined. This understanding of domain and range is fundamental in mathematics. It helps us identify the limitations of a function and ensure that we are only working with valid inputs. The concept of domain extends to various types of functions, including polynomial, trigonometric, and exponential functions, each with its own specific restrictions. By grasping the domain of a function, we gain a deeper understanding of its behavior and applicability.

While h(30) is undefined in the real number system, it does have a value in the realm of complex numbers. Complex numbers extend the number system to include the imaginary unit i, which is defined as the square root of -1: i = √(-1). Using this concept, we can express the square root of -25 as:

√(-25) = √(25 * -1) = √(25) * √(-1) = 5i

Therefore, in the complex number system, h(30) = 5i. This exploration into complex numbers provides a glimpse into a broader mathematical landscape. Complex numbers are not just abstract concepts; they have practical applications in various fields, including electrical engineering, quantum mechanics, and fluid dynamics. Understanding complex numbers opens up new avenues for solving problems that are impossible to solve using real numbers alone. This optional exploration demonstrates the interconnectedness of mathematical concepts and the power of extending our knowledge beyond the familiar. While the primary focus of this discussion is the real number system, acknowledging the existence and applicability of complex numbers enriches our mathematical understanding.

In conclusion, by carefully evaluating the function h(t) = √(5 - t) at t = 30, we have determined that h(30) is undefined in the real number system. This outcome underscores the importance of understanding the domain of a function and the limitations of the real number system. While the square root of a negative number is not a real number, it does have a value in the complex number system, where it is expressed using the imaginary unit i. This exploration has not only provided us with the answer to the specific question but has also reinforced fundamental mathematical concepts such as function evaluation, order of operations, domain, and the distinction between real and complex numbers. By breaking down the problem into manageable steps and carefully considering the implications of each step, we have gained a deeper understanding of the function and its behavior. This understanding empowers us to tackle similar mathematical challenges with confidence and precision. The journey through this problem has highlighted the beauty and interconnectedness of mathematics, demonstrating how seemingly simple questions can lead to profound insights.