Evaluating H(x) = √(-2x + 5) At X = -4.5 A Step-by-Step Guide

Introduction

In mathematics, functions are essential tools for modeling relationships between variables. One common task is to evaluate a function at a specific point, which means finding the output of the function when a particular input value is given. This article delves into the process of evaluating the function h(x) = √(-2x + 5) at x = -4.5. We'll explore the steps involved in substituting the value of x, simplifying the expression, and arriving at the final result. Understanding function evaluation is crucial for various mathematical applications, including graphing functions, solving equations, and analyzing real-world scenarios.

Understanding the Function h(x)

Before we dive into the evaluation, let's take a closer look at the function h(x) = √(-2x + 5). This function involves a square root, which means we need to ensure that the expression inside the square root (the radicand) is non-negative. In other words, -2x + 5 must be greater than or equal to 0. This constraint defines the domain of the function, which is the set of all possible input values (x) for which the function is defined. The function takes an input value x, multiplies it by -2, adds 5, and then takes the square root of the result. The output of the function, denoted by h(x), represents the value of the function at the given input x.

Step-by-Step Evaluation of h(-4.5)

Now, let's proceed with evaluating h(-4.5). This means we need to substitute x = -4.5 into the function's expression and simplify the result. Here's the step-by-step process:

1. Substitution: Replace x with -4.5 in the function's expression:

   h(-4.5) = √(-2(-4.5) + 5)

2. Multiplication: Multiply -2 by -4.5:

   h(-4.5) = √(9 + 5)

3. Addition: Add 9 and 5:

   h(-4.5) = √14

4. Square Root: Calculate the square root of 14. Since 14 is not a perfect square, we can leave the answer in radical form or approximate it using a calculator:

   h(-4.5) ≈ 3.74

Therefore, h(-4.5) = √14 ≈ 3.74. This means that when the input to the function h(x) is -4.5, the output is approximately 3.74.

Significance of the Result

The result h(-4.5) ≈ 3.74 provides a specific point on the graph of the function h(x). It tells us that the point (-4.5, 3.74) lies on the curve representing the function. This information can be useful for various purposes, such as graphing the function, finding its intercepts, or analyzing its behavior. Furthermore, this evaluation demonstrates the application of function notation and the process of substituting values into mathematical expressions. Function evaluation is a fundamental skill in algebra and calculus, and it plays a vital role in solving problems involving functions.

Importance of Function Evaluation

In mathematics, function evaluation is a fundamental skill with wide-ranging applications. It forms the basis for understanding function behavior, graphing functions, and solving equations. By evaluating functions at specific points, we gain valuable insights into their characteristics and how they relate input values to output values. This knowledge is essential for modeling real-world phenomena, making predictions, and solving practical problems.

Function Evaluation in Graphing

One of the primary uses of function evaluation is in graphing functions. To plot the graph of a function, we typically choose a set of input values (x), evaluate the function at those values to obtain the corresponding output values (h(x)), and then plot the points (x, h(x)) on a coordinate plane. By connecting these points, we can visualize the shape and behavior of the function. Evaluating functions at multiple points allows us to create a more accurate and detailed graph, revealing key features such as intercepts, maximum and minimum values, and intervals of increase and decrease. For example, to graph the function h(x) = √(-2x + 5), we could evaluate it at several x-values, such as -5, -2, 0, and 2, and then plot the resulting points. This would give us a visual representation of the function's curve.

Solving Equations and Inequalities

Function evaluation also plays a crucial role in solving equations and inequalities involving functions. When we want to find the solutions to an equation of the form h(x) = k, where k is a constant, we are essentially looking for the x-values that make the function h(x) equal to k. One way to approach this is to graph the function h(x) and the horizontal line y = k, and then find the points of intersection. The x-coordinates of these intersection points represent the solutions to the equation. Alternatively, we can use algebraic techniques to solve for x directly. In either case, function evaluation is necessary to verify the solutions and ensure that they satisfy the original equation. Similarly, function evaluation can be used to solve inequalities involving functions, such as h(x) > k or h(x) < k.

Modeling Real-World Scenarios

Many real-world phenomena can be modeled using functions. For example, the population of a city over time, the trajectory of a projectile, or the growth of a plant can all be represented by mathematical functions. In these cases, function evaluation allows us to make predictions and answer questions about the modeled scenario. By evaluating the function at a specific time, we can estimate the population size, the projectile's height, or the plant's size. Function evaluation provides a powerful tool for analyzing and understanding the behavior of real-world systems.

Domain Considerations for h(x) = √(-2x + 5)

As mentioned earlier, the function h(x) = √(-2x + 5) involves a square root, which imposes a restriction on the possible input values. The radicand, -2x + 5, must be non-negative for the function to be defined. This leads to the inequality:

-2x + 5 ≥ 0

To solve this inequality, we can follow these steps:

1. Subtract 5 from both sides:

   -2x ≥ -5

2. Divide both sides by -2 (and reverse the inequality sign because we are dividing by a negative number):

   x ≤ 5/2

Therefore, the domain of the function h(x) = √(-2x + 5) is the set of all real numbers x such that x ≤ 5/2. This means that the function is only defined for input values that are less than or equal to 2.5. When evaluating the function, it's essential to keep the domain in mind to avoid undefined results. If we try to evaluate the function at an x-value greater than 2.5, the radicand will be negative, and the square root will be undefined in the real number system.

Impact on Evaluation

The domain restriction affects the evaluation of the function because we can only evaluate it at x-values within the domain. In our example, we evaluated h(-4.5), and since -4.5 ≤ 5/2, this evaluation was valid. However, if we were to try to evaluate h(3), we would find that -2(3) + 5 = -1, which is negative. Therefore, h(3) is undefined, and 3 is not in the domain of the function. Understanding the domain of a function is crucial for interpreting the results of function evaluation and ensuring that they are meaningful.

Alternative Approaches to Evaluation

While we evaluated h(-4.5) by direct substitution, there are alternative approaches that can be used in certain situations. These approaches may be more efficient or provide additional insights into the function's behavior.

Graphical Evaluation

If we have the graph of the function h(x), we can evaluate it graphically by finding the point on the graph that corresponds to the input value x = -4.5. The y-coordinate of that point will be the value of h(-4.5). This method can be particularly useful when we have a visual representation of the function and want to quickly estimate its value at a given point. However, graphical evaluation may not always be precise, especially if the graph is not drawn accurately or if the point of interest falls between grid lines.

Numerical Approximation

In some cases, we may not be able to find an exact value for the function, or we may only need an approximate value. In such situations, we can use numerical methods to approximate the function's value. One common method is to use a calculator to directly compute the value of the function. For example, to approximate h(-4.5), we can use a calculator to find the square root of 14, which gives us approximately 3.74. Numerical approximation can be useful when dealing with complex functions or when an exact value is not required.

Conclusion

In conclusion, evaluating the function h(x) = √(-2x + 5) at x = -4.5 involves substituting the value of x into the function's expression, simplifying the result, and considering the domain of the function. We found that h(-4.5) = √14 ≈ 3.74. Function evaluation is a fundamental skill in mathematics, with applications in graphing, solving equations, and modeling real-world scenarios. By understanding the process of function evaluation and its significance, we can gain a deeper understanding of mathematical functions and their role in various contexts.