Find H(x) = F(x) * G(x). Easy Math Solution!

by Andrew McMorgan 45 views

Hey Plastik Magazine readers! Today, we're diving into a fun little math problem that's super useful for understanding functions. We've got two functions, f(x) = 5x and g(x) = x + 11, and our mission is to find a new function h(x) that's the product of f(x) and g(x). Sounds like a plan? Let's jump right in!

Understanding the Basics: What are Functions?

Before we start solving, let's quickly recap what functions are all about. In simple terms, a function is like a machine: you feed it an input (usually denoted as x), and it spits out an output based on a specific rule. Our f(x) takes any input x and multiplies it by 5. So, if you put in 2, it gives you 10. Our g(x) takes any input x and adds 11 to it. Simple, right?

Why are functions important, you ask? Well, they're the building blocks of many mathematical models and real-world applications. From predicting stock prices to designing bridges, functions help us understand and manipulate relationships between different variables. So, mastering this stuff is totally worth it!

Step-by-Step: Finding h(x)

Alright, let's get our hands dirty and find h(x). Remember, h(x) = f(x) * g(x). This means we need to multiply the expressions for f(x) and g(x) together.

Here's how we do it:

  1. Write down the expressions:
    • f(x) = 5x
    • g(x) = x + 11
  2. Multiply them together:
    • h(x) = (5x) * (x + 11)
  3. Use the distributive property: This is where we multiply 5x by both x and 11.
    • h(x) = (5x * x) + (5x * 11)
  4. Simplify:
    • h(x) = 5x² + 55x

And there you have it! h(x) = 5x² + 55x. That wasn't so bad, was it?

Why is the distributive property so important? It's a fundamental rule in algebra that allows us to multiply a single term by a group of terms inside parentheses. Without it, we couldn't properly combine f(x) and g(x) to find h(x).

Checking the Options

Now that we've found h(x), let's compare it to the options given:

A. h(x) = 5x + 55

B. h(x) = 5x + 55x

C. h(x) = 5x² + 55x

D. h(x) = 5x² + 55

It's clear that option C, h(x) = 5x² + 55x, matches our solution perfectly. So, that's the correct answer!

Why the Other Options are Wrong

Just for kicks, let's quickly look at why the other options don't work:

  • Option A: h(x) = 5x + 55

    • This looks like someone just added f(x) and a constant related to g(x), instead of multiplying the two functions. No distribution here!

  • Option B: h(x) = 5x + 55x

    • This option adds 5x and 55x, which is not the correct way to multiply the functions. It seems like a mix-up of addition and multiplication.

  • Option D: h(x) = 5x² + 55

    • This one has the 5x² term, but it's missing the 55x term. It seems like the multiplication by 11 was forgotten.

Understanding why these options are wrong helps reinforce the correct method and avoid common mistakes.

Real-World Applications

Okay, so we solved a math problem. But where does this stuff actually come in handy? Let's think about a business scenario. Suppose f(x) = 5x represents the cost of producing x items, and g(x) = x + 11 represents the selling price per item (a base price plus a markup). Then, h(x) = f(x) * g(x) = 5x² + 55x would represent the total revenue from selling x items. Knowing this, the business can analyze how revenue changes with the number of items produced and sold.

Another example could be in physics. Imagine f(x) represents the force applied to an object, and g(x) represents the distance the object moves. Then, h(x) could represent the work done, which is the product of force and distance. These concepts pop up everywhere, showing just how versatile functions can be.

Practice Makes Perfect

Want to get even better at this? Here are a few practice problems you can try:

  1. If f(x) = 2x and g(x) = x - 3, find h(x) = f(x) * g(x).
  2. If f(x) = x + 5 and g(x) = 3x, find h(x) = f(x) * g(x).
  3. If f(x) = 4x and g(x) = 2x + 1, find h(x) = f(x) * g(x).

The more you practice, the easier this will become. And remember, math is like building with LEGOs: each concept builds upon the previous one, so keep stacking those blocks!

Conclusion: You've Got This!

So, there you have it! We've successfully found h(x) = f(x) * g(x), understood why the other options were incorrect, and even explored some real-world applications. You're now one step closer to becoming a function master!

Remember, math isn't about memorizing formulas; it's about understanding the underlying concepts and applying them creatively. Keep practicing, stay curious, and don't be afraid to ask questions. You've got this!

Thanks for hanging out with us at Plastik Magazine. Keep an eye out for more math adventures in the future. Until next time, stay stylish and keep those functions in check!