Finding (fg)(x) And Evaluating At -2: A Step-by-Step Guide

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Hey math enthusiasts! Today, we're diving into the world of function composition and evaluation. We're given three functions: f(x), g(x), and h(x). Our mission, should we choose to accept it, is to find the composite function (fg)(x) and then evaluate it at x = -2. Don't worry, it's not as scary as it sounds! Let's break it down step-by-step and make sure you've got a solid understanding of this concept. Ready to roll, guys?

Understanding Function Composition

Function composition is all about combining functions. When we see something like (fg)(x), it means we're taking the function g(x) and plugging it into the function f(x). Think of it like a mathematical nesting doll. We're essentially replacing the 'x' in f(x) with the entire expression of g(x). This creates a new function that represents the combined action of both f and g. The notation (fg)(x) is the same as f(g(x)), so you might see it written either way. They both mean the same thing.

Now, let's get down to the nitty-gritty. We're given:

  • f(x) = 2x^2 + 2x - 3
  • g(x) = 3x + 4
  • h(x) = 3x^2 - 1 (We won't need h(x) for this particular problem, but it's there in case you need it later!)

Our goal is to find (fg)(x). This means we need to find f(g(x)). So, we'll take the entire expression for g(x) which is 3x + 4 and plug it in for every 'x' that appears in the function f(x). This might seem a little abstract at first, but with practice, it'll become second nature.

So, instead of f(x) = 2x^2 + 2x - 3, we will have f(g(x)) = 2(3x + 4)^2 + 2(3x + 4) - 3. See how we replaced every 'x' in f(x) with the entire expression (3x + 4)? Now all that's left is simplifying the expression. It's essentially an algebra problem, so don't be afraid to take it step by step. We'll start with the squared term using the formula (a+b)^2 = a^2 + 2ab + b^2 and then work our way through simplifying the whole expression.

Step-by-Step Calculation of (fg)(x)

Okay, guys, let's get our hands dirty and actually calculate (fg)(x). As we mentioned before, we're going to substitute g(x) into f(x). Here's the breakdown of each step, and don't worry, it's not rocket science!

  1. Substitute g(x) into f(x):

    We begin with f(x) = 2x^2 + 2x - 3. Replace every 'x' with (3x + 4): f(g(x)) = 2(3x + 4)^2 + 2(3x + 4) - 3.

  2. Expand the squared term:

    Remember that (3x + 4)^2 is the same as (3x + 4)(3x + 4). Expanding this gives us 9x^2 + 24x + 16. So, our expression becomes:

    2(9x^2 + 24x + 16) + 2(3x + 4) - 3

  3. Distribute the constants:

    Multiply the '2' across the first set of parentheses and the other '2' across the second set:

    18x^2 + 48x + 32 + 6x + 8 - 3

  4. Combine like terms:

    Identify and combine the terms with the same power of 'x' (i.e., the x^2 terms, the x terms, and the constant terms):

    18x^2 + (48x + 6x) + (32 + 8 - 3)

  5. Simplify:

    Combining the terms, we get:

    (fg)(x) = 18x^2 + 54x + 37

    Voilà! We've successfully found (fg)(x). This new quadratic function represents the composition of f and g.

Evaluating (fg)(-2)

Now that we've found the formula for (fg)(x), we can easily evaluate it at x = -2. Evaluating a function at a specific value means substituting that value for 'x' in the function's formula. In other words, you have already done most of the work to get the result.

Here’s how we'll do it:

  1. Substitute x = -2 into (fg)(x):

    We know that (fg)(x) = 18x^2 + 54x + 37. So, (fg)(-2) = 18(-2)^2 + 54(-2) + 37.

  2. Calculate the powers:

    Remember to follow the order of operations (PEMDAS/BODMAS): calculate the exponent first. (-2)^2 = 4, so our expression becomes:

    (fg)(-2) = 18(4) + 54(-2) + 37

  3. Perform the multiplications:

    18 * 4 = 72 and 54 * -2 = -108, giving us:

    (fg)(-2) = 72 - 108 + 37

  4. Combine the terms:

    72 - 108 + 37 = 1

    Therefore, (fg)(-2) = 1. This is the final answer! This means that when we combine the functions f and g and then plug in -2, the result is 1. That's a very specific value and gives us valuable information about the behavior of these combined functions at a specific point.

Summary and Key Takeaways

Function composition is a fundamental concept in mathematics. Today, we've walked through the steps required to find and evaluate a composite function. Let's recap what we did, so you're crystal clear on the process. In the beginning, we learned what function composition means; we substitute one function into the other. Then, we calculated (fg)(x) by substituting g(x) into f(x) and simplifying the result. Finally, we evaluated (fg)(-2) by substituting -2 for x in the simplified expression of (fg)(x). The entire process might seem a bit lengthy when explained step by step, but with practice, you can easily tackle such problems.

Here are the key takeaways:

  • (fg)(x) means f(g(x)): Substitute the entire expression of g(x) into f(x).
  • Simplify: Expand and simplify the resulting expression by combining like terms.
  • Evaluate: Substitute the given value of 'x' into the simplified composite function to find the final result.

These principles apply across various types of functions. So, whether you're dealing with polynomials, trigonometric functions, or exponential functions, the core process of function composition remains the same. The only difference is the specific algebraic manipulations that you'll have to use. The more you work with function composition, the easier and more intuitive it becomes. Remember, practice is key! Try working through more examples to reinforce your understanding. Keep at it, and you'll be composing functions like a pro in no time! Have fun exploring the exciting world of functions, and keep up the great work!