Evaluate H(-8): A Step-by-Step Solution

by Rajiv Sharma 40 views

Hey guys! Today, we're diving into a fun little math problem where we need to evaluate a function. Specifically, we've got a function h(x), and our mission, should we choose to accept it, is to find out what h(-8) is. In simpler terms, we need to plug in -8 wherever we see an x in our function and then do the math. Sounds like a plan? Let's break it down step by step. So let's get started and make math a little less intimidating and a lot more fun!

Understanding the Function

Before we jump into plugging in numbers, let's first make sure we understand what the function h(x) is all about. We're given:

h(x) = (x^2 + 3x) / (4x + 27)

This might look a bit intimidating at first glance, but don't worry, it's just a recipe! Think of h(x) as a machine. You feed it a number (x), and it spits out another number based on the instructions in the formula. The formula here tells us to:

  1. Square the input (x), which means multiply it by itself.
  2. Multiply the input (x) by 3.
  3. Add the results from steps 1 and 2.
  4. Multiply the input (x) by 4.
  5. Add 27 to the result from step 4.
  6. Finally, divide the result from step 3 by the result from step 5.

That's all there is to it! Now that we understand what the function h(x) does, we're ready to feed it the number -8 and see what comes out. Remember, the key to understanding functions is to see them as a set of instructions. Once you grasp the instructions, evaluating becomes much easier. Functions are a fundamental concept in mathematics, and they pop up everywhere, from simple algebra to complex calculus. So getting comfortable with them now will definitely pay off in the long run. We're essentially building the groundwork for more advanced mathematical adventures. So, keep this idea of a function as a set of instructions in mind as we move forward. It will help you visualize and tackle more complicated problems down the road.

Plugging in -8 for x

Alright, now for the fun part: plugging in -8 for x in our function. This is where the rubber meets the road, guys! We're going to take the recipe we discussed earlier and actually use it. So, wherever we see an x, we're going to replace it with -8. Let's write it out:

h(-8) = ((-8)^2 + 3(-8)) / (4(-8) + 27)

Notice how we've carefully substituted -8 for each x. It's super important to pay attention to the signs (positive and negative) here, as a small mistake can throw off the whole calculation. We've also used parentheses around the -8 to make it clear that the negative sign is part of the number we're squaring and multiplying. This is a good practice to avoid confusion, especially when dealing with negative numbers. Now, we have a numerical expression that we need to simplify. This involves following the order of operations, which you might remember as PEMDAS or BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). We'll tackle the exponents first, then the multiplications, then the additions, and finally the division. By following this order, we ensure that we get the correct answer. So, let's roll up our sleeves and start simplifying this expression. We're on our way to finding the value of h(-8)!

Step-by-Step Calculation

Okay, let's break down the calculation step by step, making sure we don't miss anything. Remember, it's all about following the order of operations (PEMDAS/BODMAS). First up, we've got the exponents:

(-8)^2 = (-8) * (-8) = 64

A negative number multiplied by a negative number gives us a positive number. So, (-8) squared is 64. Now, let's deal with the multiplications:

3(-8) = -24
4(-8) = -32

We've got 3 times -8, which is -24, and 4 times -8, which is -32. Now, let's substitute these values back into our expression:

h(-8) = (64 + (-24)) / (-32 + 27)

Next, we perform the additions inside the parentheses:

64 + (-24) = 64 - 24 = 40
-32 + 27 = -5

So, we've simplified the numerator to 40 and the denominator to -5. Now our expression looks much simpler:

h(-8) = 40 / (-5)

Finally, we perform the division:

40 / (-5) = -8

And there you have it! We've calculated that h(-8) = -8. Each step in this calculation is crucial, and understanding why we perform them in this order is key to mastering mathematical expressions. It's like building a house; you need a solid foundation (understanding the order of operations) before you can start adding the walls and roof.

The Final Answer

So, after all that careful calculation, we've arrived at our final answer. Drumroll, please…

h(-8) = -8

That's it! When we plug -8 into our function h(x), the output is -8. It's pretty cool how a function can take one number and transform it into another, isn't it? This result tells us a specific point on the graph of the function h(x). If we were to plot the graph, the point (-8, -8) would lie on the curve. This is a fundamental concept in algebra and calculus – connecting functions to their graphical representations. Understanding this connection allows us to visualize the behavior of functions and solve a wide range of problems. So, remember, h(-8) = -8 is not just a numerical answer; it's a piece of information about the function h(x) and its graph. And that's what makes math so fascinating – it's all interconnected!

Key Takeaways

Let's quickly recap the key takeaways from this exercise. First and foremost, we learned how to evaluate a function for a specific input. This involves substituting the given value for the variable (x in this case) and then simplifying the expression using the order of operations. We also emphasized the importance of paying attention to signs, especially when dealing with negative numbers. A small sign error can lead to a completely different answer. Furthermore, we reinforced the concept of the order of operations (PEMDAS/BODMAS) as the golden rule for simplifying mathematical expressions. By following this order, we ensure that our calculations are accurate and consistent. Finally, we highlighted the connection between a function's output and its graphical representation, showing how h(-8) = -8 corresponds to a point on the graph of h(x). These takeaways are not just specific to this problem; they are fundamental concepts that will help you tackle a wide range of mathematical challenges. So, keep these principles in mind, and you'll be well-equipped to conquer more complex problems in the future. Remember, practice makes perfect, so keep evaluating those functions!

I hope you found this explanation helpful and easy to follow! Evaluating functions is a core skill in mathematics, and mastering it opens doors to more advanced concepts. If you have any questions or want to try another example, feel free to ask. Happy calculating!