Solving Composite Functions: Step-by-Step Math Guide

by ADMIN 53 views

Hey guys! Today, we're diving deep into the world of composite functions. This might sound intimidating, but trust me, it's super manageable once you break it down. We're going to tackle some common problems step-by-step, so you can ace those math quizzes and exams. Let's get started!

1. Finding the Value of 'p' in a Composite Function

Understanding the Problem:

Our first problem involves finding the value of 'p' given two functions, f(x) = 2x - 3 and g(x) = 3x + 5, and the composite function (f o g)(p) = 90. What does this all mean? Well, (f o g)(p) means we're plugging the function g(x) into f(x), and then substituting 'p' for x. The goal is to find the value of 'p' that makes the whole expression equal to 90. Let's break it down into smaller steps to make it crystal clear.

Step 1: Calculate (f o g)(x)

The first thing we need to do is figure out what (f o g)(x) actually looks like. Remember, this means we're plugging g(x) into f(x). So, wherever we see an x in f(x), we're going to replace it with the entire function g(x). Sounds fun, right?

  • We start with f(x) = 2x - 3.
  • Now, we replace x with g(x), which is 3x + 5. This gives us f(g(x)) = 2(3x + 5) - 3.

Step 2: Simplify the Expression

Next, let's simplify the expression we just got. This involves distributing the 2 and combining like terms. Don't worry, it's just basic algebra!

  • We have 2(3x + 5) - 3.
  • Distribute the 2: 6x + 10 - 3.
  • Combine the constants: 6x + 7. So, (f o g)(x) = 6x + 7.

Step 3: Substitute 'p' and Set up the Equation

Now that we have (f o g)(x) = 6x + 7, we can substitute 'p' for x. This gives us (f o g)(p) = 6p + 7. We know that (f o g)(p) = 90, so we can set up the equation: 6p + 7 = 90. We're getting closer to solving for 'p'!

Step 4: Solve for 'p'

Alright, time to solve for 'p'. This is just a simple algebraic equation. We need to isolate 'p' on one side of the equation.

  • Start with 6p + 7 = 90.
  • Subtract 7 from both sides: 6p = 83.
  • Divide both sides by 6: p = 83/6.

Step 5: The Answer

And there we have it! The value of p is 83/6. You might see this written as a fraction or a decimal, but either way, we've successfully found the answer. Remember, the key here was breaking down the problem into smaller, manageable steps. We first found the composite function, then substituted, and finally solved for the unknown.

2. Evaluating (h o g o f) (2)

Understanding the Problem

Okay, let's tackle another composite function problem, but this time with three functions! We're given f(x) = 2x + 3, g(x) = x - 1, and h(x) = x². Our mission, should we choose to accept it, is to find the value of (h o g o f)(2). This might look like alphabet soup, but don't worry, we'll break it down. The notation (h o g o f)(2) means we're plugging f(2) into g, and then plugging the result into h. It's like a mathematical assembly line!

Step 1: Evaluate f(2)

The first step is to find the value of f(2). This means we're simply substituting x = 2 into the function f(x). Let's do it:

  • We have f(x) = 2x + 3.
  • Substitute x = 2: f(2) = 2(2) + 3.
  • Simplify: f(2) = 4 + 3 = 7. So, f(2) = 7. Easy peasy!

Step 2: Evaluate g(f(2))

Now that we know f(2) = 7, we can plug this value into the function g(x). So we're finding g(7). Remember, g(x) = x - 1. Let's plug it in:

  • We have g(x) = x - 1.
  • Substitute x = 7: g(7) = 7 - 1.
  • Simplify: g(7) = 6. So, g(f(2)) = 6. We're making progress!

Step 3: Evaluate h(g(f(2)))

We're almost there! We know that g(f(2)) = 6. Now we need to plug this value into the function h(x). So we're finding h(6). We know that h(x) = x². Let's do the math:

  • We have h(x) = x².
  • Substitute x = 6: h(6) = 6².
  • Simplify: h(6) = 36. So, h(g(f(2))) = 36.

Step 4: The Answer

And that's it! The value of (h o g o f)(2) is 36. We successfully worked our way through the composite function, one step at a time. Remember, the key is to start from the inside and work your way outwards. We evaluated f(2) first, then used that result to evaluate g, and finally used that result to evaluate h.

3. Solving for an Unknown in Composite Functions

Understanding the Problem

Okay, let's crank up the challenge a bit. This time, we're given that g(f(x)) = f(g(x)). This means the composite functions in both orders are equal. We also know that f(x) = 3x + p and g(x) = 2x + .... Notice the little dots there? That's our mystery! We need to find the complete expression for g(x) and then solve for any unknowns. This is a classic problem that tests your understanding of composite functions and algebraic manipulation.

Step 1: Write the complete expression

Let's assume the complete expression for g(x) = 2x + q. This will make it easier to follow the next calculations.

Step 2: Find g(f(x))

First, let's find the expression for g(f(x)). This means we're plugging f(x) into g(x). So, wherever we see an x in g(x), we'll replace it with 3x + p.

  • We have g(x) = 2x + q.
  • Replace x with f(x) = 3x + p: g(f(x)) = 2(3x + p) + q.
  • Simplify: g(f(x)) = 6x + 2p + q.

Step 3: Find f(g(x))

Next, let's find the expression for f(g(x)). This means we're plugging g(x) into f(x). So, wherever we see an x in f(x), we'll replace it with 2x + q.

  • We have f(x) = 3x + p.
  • Replace x with g(x) = 2x + q: f(g(x)) = 3(2x + q) + p.
  • Simplify: f(g(x)) = 6x + 3q + p.

Step 4: Set g(f(x)) equal to f(g(x))

We're given that g(f(x)) = f(g(x)). So, we can set the expressions we found in steps 2 and 3 equal to each other:

  • 6x + 2p + q = 6x + 3q + p

Step 5: Solve for the Unknown

Now, we need to solve for the unknown, which in this case is the relationship between p and q. Notice that the 6x terms cancel out on both sides, which is a good sign. Let's simplify and solve:

  • Subtract 6x from both sides: 2p + q = 3q + p.
  • Subtract p from both sides: p + q = 3q.
  • Subtract q from both sides: p = 2q.

Step 6: The Answer

So, we've found the relationship between p and q: p = 2q. This means that p is twice the value of q. Depending on the specific problem, you might be given a value for p or q and asked to find the other. Or, you might just be asked to find the relationship between them, which we've done here. The key was to carefully find the composite functions in both orders, set them equal to each other, and then use algebra to solve for the unknown.

Conclusion

And there you have it! We've tackled some challenging composite function problems today. Remember, the key to success with composite functions is to break them down into smaller, manageable steps. Start from the inside and work your way outwards, carefully substituting and simplifying as you go. With a little practice, you'll be a composite function pro in no time! Keep practicing, and you'll conquer those math challenges. You got this!