Calculating Function Values: A Step-by-Step Guide
Hey there, math enthusiasts! Today, we're diving into the world of functions and figuring out how to find their values at specific points. We'll be working with a function, h(x) = √( (x - 2)² ), and our mission is to calculate h(8), h(3), h(0), and h(-5). Don't worry if this sounds a bit intimidating; we'll break it down step by step, making it super easy to understand. Let's get started!
Understanding the Function and the Task
First things first, let's make sure we're all on the same page. The function h(x) = √( (x - 2)²) is a mathematical rule that takes an input (which we call 'x') and transforms it into an output. In this case, the function does the following: It subtracts 2 from the input 'x', squares the result, and then takes the square root of the final value. Our task is to find the output of this function when we plug in different values for 'x': 8, 3, 0, and -5. Essentially, we are replacing 'x' in the function's formula with each of these numbers, one at a time, and simplifying the expression to find the final result.
So, why is this important? Well, understanding how to evaluate functions is a fundamental skill in mathematics. It's like having a key to unlock a variety of mathematical problems and concepts. Functions are used everywhere, from calculating the trajectory of a ball to modeling economic growth. Being able to evaluate a function quickly and accurately is an essential skill. Furthermore, this specific function gives us a chance to brush up on our knowledge of square roots and absolute values. When dealing with square roots and squares, it is crucial to remember a key point: the square root of a square of any number is the absolute value of that number. For example, the square root of (-3)² is not -3, but 3. Understanding this concept will be useful when we begin to solve for the values. We can also test our understanding of how negative values can interact with mathematical functions. By learning this skill, you're not just doing a math problem; you're building a foundation for more advanced topics. In fact, understanding functions is key to understanding calculus. Functions are also used in computer programming to create reusable pieces of code and perform calculations.
Calculating h(8)
Alright, let's start with h(8). This means we need to replace 'x' with 8 in our function: h(8) = √((8 - 2)²). Let's break it down further. First, we need to solve what's inside the parentheses: 8 - 2 = 6. Our equation now looks like this: h(8) = √(6²). Next, we need to square 6: 6² = 36. Now, the equation is: h(8) = √36. Finally, we calculate the square root of 36, which is 6. So, h(8) = 6. This means that when we input 8 into the function, the output is 6. Great job, you did it!
Let's clarify what's going on here. We started with the function definition and replaced the variable 'x' with the value 8. Then we carefully followed the order of operations, simplifying the equation step by step. We first calculated the value inside the parenthesis (8 - 2). Next, we squared the result (6²). Finally, we took the square root of that result. The whole point here is to become comfortable with the process of evaluating a function. Remember that function notation, like h(8), tells us exactly what to do: plug in the value inside the parenthesis and follow the function's instructions. In summary, evaluating h(8) involved plugging in the value, following the order of operations, and arriving at the final answer. This process helps us not only find the correct answer, but also build a strong understanding of how functions work. Don't worry if it seems difficult at first, practice makes perfect! Soon enough, these calculations will become second nature.
Calculating h(3)
Now, let's calculate h(3). We replace 'x' with 3 in our function: h(3) = √((3 - 2)²). Let's start simplifying within the parentheses: 3 - 2 = 1. Our equation becomes: h(3) = √(1²). Next, we square 1: 1² = 1. Our equation is now: h(3) = √1. Finally, the square root of 1 is 1. Thus, h(3) = 1. This indicates that when we put 3 into the function, the output we receive is 1. Amazing work!
We again went through the same process, but this time with a different input. Notice how the steps remain consistent: replace the variable, simplify inside parentheses, square the result, and take the square root. The importance of the order of operations becomes clear here, because it ensures that we perform the calculations in the correct sequence. The steps must be followed methodically, one by one. This is also a good opportunity to review your knowledge of square roots and squares. In the end, functions are all about the relationship between input and output, and this process helps you better understand that relationship. Keep in mind that as you continue to work with different functions, the specific calculations will change. However, the fundamental process of replacing the variable with the input value and simplifying remains the same. Remember, practice is key, and each calculation brings you one step closer to mastering functions. Always remember to take it step by step, and don't be afraid to double-check your work to catch any potential errors. Keep up the great effort, and you'll be a function whiz in no time!
Calculating h(0)
Next up, we need to calculate h(0). We plug in 0 for 'x' in our function: h(0) = √((0 - 2)²). First, let's solve the parentheses: 0 - 2 = -2. Our equation is now: h(0) = √((-2)²). Next, we square -2: (-2)² = 4. The equation becomes: h(0) = √4. Finally, the square root of 4 is 2. So, h(0) = 2. Excellent work, guys! You're doing a fantastic job.
Let's pause to reflect on our progress. Notice that in this step, we introduced a negative number into the calculation. Remember that when you square a negative number, the result is always positive. This is why (-2)² equals 4, not -4. This is a very common point of confusion, so be sure you understand it. It also underscores the importance of the order of operations. Without following them, you might accidentally arrive at an incorrect answer. Furthermore, note that the function transforms its inputs in a predictable way. By observing these changes, you can better understand how the function works. The function first subtracts 2, then squares the result, and finally, takes the square root. Each of these steps plays a crucial role in determining the final output. In the future, you may see functions with more complex calculations; but the basic principle of substituting the value for 'x' and following the instructions of the function will still remain the same. The more you practice, the easier it will become to recognize patterns and efficiently solve these types of problems. You are now familiar with calculating h(0).
Calculating h(-5)
Finally, let's calculate h(-5). We replace 'x' with -5: h(-5) = √((-5 - 2)²). First, solve inside the parentheses: -5 - 2 = -7. Our equation is now: h(-5) = √((-7)²). Next, we square -7: (-7)² = 49. So, the equation becomes: h(-5) = √49. Finally, the square root of 49 is 7. Therefore, h(-5) = 7. Awesome job!
Here we went through another example of dealing with negative numbers. Just like before, the square of the negative number became positive. As you get more used to these kinds of problems, you may see that some functions have special properties. In this case, our function, after simplification, is actually equal to the absolute value of x - 2. We can show this by first simplifying √( (x - 2)²) to |x - 2|. If you didn't know, the absolute value of a number is its distance from zero, so it is always positive. This can often simplify the process. For example, h(8) is equivalent to |8 - 2|, which equals |6|, or 6. Likewise, h(-5) is equivalent to |-5 - 2|, which equals |-7|, or 7. By identifying these patterns, you can gain a deeper understanding of how the function operates and recognize quicker ways to solve the problem. As you learn more about different kinds of functions, you'll be able to spot these kinds of patterns more easily and efficiently solve them.
Summary of Results
Alright, let's summarize our findings:
h(8) = 6h(3) = 1h(0) = 2h(-5) = 7
We successfully found the function values for each given input. Great work, everyone!
Conclusion: Mastering Function Values
Congratulations! You've successfully navigated the process of calculating function values. Remember, this is a fundamental skill in mathematics. We started with a function, substituted different values for the variable 'x', and carefully followed the order of operations to find the outputs. You've now gained a solid understanding of how to evaluate functions, from plugging in values to simplifying expressions. This is a crucial skill for more advanced math topics. Keep practicing and exploring different functions. The more you practice, the more comfortable you'll become with this essential mathematical concept. Keep up the excellent work! You are now well-equipped to tackle a wide variety of function problems.