Calculating Function Values: F, G, And H Explained

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Hey everyone! Today, we're diving into the world of functions. We'll be working with three specific functions: f, g, and h. Don't worry, it's not as scary as it sounds! We'll go through the calculations step by step, so you'll totally get the hang of it. We're going to find the values of these functions at specific points. Ready to jump in? Let's go!

Understanding the Functions: A Quick Overview

First off, let's get acquainted with our functions. We've got:

  • f(x)=∣6x∣−11f(x) = |6x| - 11
  • g(x)=−2x+13g(x) = \sqrt{-2x + 13}
  • h(x)=xx2−7h(x) = \frac{x}{x^2 - 7}

Each of these is a rule that takes an input (which we call x) and spits out an output. Think of it like a machine: you put something in, and something else comes out. The rules are the instructions the machine follows. Understanding the functions is key to solving the problems, so before we begin, let's have a quick review. The f(x) function involves an absolute value. The absolute value of a number is its distance from zero, so it's always positive (or zero). The g(x) function involves a square root. Remember, you can only take the square root of a non-negative number. That's super important to keep in mind! Finally, the h(x) function is a fraction, and fractions can be a little tricky because we have to be careful about dividing by zero. Now that we have brushed up our knowledge of the functions, let us go through the calculations to get the answers.

Function f(x): Absolute Value Adventures

Let's start with f(x). The function is defined as f(x) = |6x| - 11. This means we take our input, multiply it by 6, find the absolute value, and then subtract 11. The question requires us to find f(-2/3). So, we'll substitute x with -2/3. Therefore, f(-2/3) = |6 * (-2/3)| - 11. Now let's break this down:

  1. Multiply: 6 * (-2/3) = -4.
  2. Absolute Value: |-4| = 4. Remember that the absolute value gives us the distance from zero.
  3. Subtract: 4 - 11 = -7.

So, f(-2/3) = -7. Easy peasy, right? The absolute value might seem a bit weird at first, but with practice, it becomes second nature. It's all about understanding that the absolute value symbol makes everything positive (unless it's zero).

Function g(x): Rooting Out the Answer

Next up, we have g(x) = \sqrt{-2x + 13}. This one involves a square root, so we need to be careful that the expression inside the square root is not negative. We're asked to find g(2). Let's plug in x = 2 and see what we get:

  1. Substitute: g(2) = \sqrt{-2(2) + 13}
  2. Multiply: -2 * 2 = -4
  3. Add: -4 + 13 = 9
  4. Square Root: \sqrt{9} = 3

So, g(2) = 3. Great job! The square root part might seem intimidating, but as long as we keep the order of operations straight, it's not so bad. We make sure to handle the multiplication and addition inside the square root before taking the square root itself. It is also important to remember that we can only take the square root of a positive number.

Function h(x): Fraction Fun

Finally, let's find h(4) using the function h(x) = \frac{x}{x^2 - 7}. This one has a fraction, so we'll need to be extra cautious. Let's substitute x with 4:

  1. Substitute: h(4) = \frac{4}{4^2 - 7}
  2. Exponent: 4^2 = 16
  3. Subtract: 16 - 7 = 9
  4. Divide: 4/9

Therefore, h(4) = 4/9. Fractions can look complicated, but following the order of operations, just like any other expression, simplifies them greatly. The important thing to keep in mind with fractions is that the denominator can't be zero. Lucky for us, in this case, it wasn't! And now you know the values of f(-2/3), g(2), and h(4). Nicely done!

Summarizing the Results

Let's recap what we've found:

  • f(-2/3) = -7
  • g(2) = 3
  • h(4) = 4/9

And that's it! You've successfully calculated the values of these functions at specific points. Awesome work! You've now gained some valuable experience working with different types of functions and performing calculations. Remember, practice makes perfect. The more you work with functions, the more comfortable and confident you'll become.

Further Exploration and Practice

Now that you understand the basics, you might be wondering, "What else can I do with functions?" Well, the possibilities are endless! Functions are fundamental in mathematics and show up in all sorts of different areas. Here are some ideas for your next steps:

  • Try Different Inputs: Plug in different values for x into the functions we discussed. See what happens! This helps you build intuition about how the functions behave.
  • Graphing Functions: Learn how to graph these functions. Seeing them visually can provide a deeper understanding of their properties, such as their domain, range, and where they increase or decrease.
  • Compose Functions: This is when you put the output of one function into another function. For example, what is f(g(2))? This adds another layer of complexity and lets you explore how functions interact.
  • Explore Different Types of Functions: There are so many types of functions out there. Quadratic, exponential, trigonometric – the list goes on! Each one has its own special rules and behaviors.
  • Real-World Applications: Think about how functions are used in the real world. From physics to finance to computer science, functions are used everywhere.

Practice Problems

To solidify your understanding, try these practice problems:

  1. Find f(0), g(6), and h(1).
  2. What is g(5)? Does it work? Why or why not?
  3. Graph f(x), g(x), and h(x).
  4. Calculate f(h(4)).

These exercises will not only reinforce what you've learned but also give you a taste of the different kinds of problems you can solve with functions. Remember, the key to success is to keep practicing and exploring! The more you engage with the material, the more it will click.

Conclusion: You've Got This!

Functions may have seemed a little daunting at first, but you've tackled them head-on, and you've done a fantastic job! By breaking down each function and following the rules, you've successfully calculated their values at specific points. Keep up the excellent work, and always remember to practice, experiment, and most importantly, have fun with math! You're building a strong foundation for future mathematical adventures. Keep exploring, keep questioning, and keep learning. Math is an amazing subject with endless possibilities. And hey, if you get stuck, don't worry! That's part of the process. Ask questions, seek help, and celebrate your progress along the way. You've totally got this!