Composite Functions Explained Step-by-Step F(x)=3x-8 And G(x)=x+9

by Sam Evans 66 views
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Hey guys! Today, we're going to explore the fascinating world of composite functions. We'll be working with two simple, yet powerful functions: f(x) = 3x - 8 and g(x) = x + 9. Our goal is to understand how these functions interact when we combine them and, most importantly, what the domains of these newly formed composite functions look like. So, buckle up and let's dive in!

Understanding Composite Functions

Before we jump into the specifics, let's quickly recap what a composite function actually is. Simply put, a composite function is a function that's formed by plugging one function into another. Think of it like a machine where you feed in an input, and the first function processes it. Then, the output of that first function becomes the input for the second function. This chain reaction is what makes composite functions so interesting.

We denote a composite function using a small circle, like this: (f ∘ g)(x). This is read as "f of g of x," and it means we first apply the function g to x, and then we apply the function f to the result. The order is crucial here, guys, because (f ∘ g)(x) is generally not the same as (g ∘ f)(x). Let's keep this in mind as we work through our examples.

Now, let's get our hands dirty and start calculating some composite functions!

a. Finding (f ∘ g)(x)

Our first task is to determine the composite function (f ∘ g)(x). Remember, this means we need to plug the function g(x) into the function f(x). Let's break this down step by step:

  1. Start with the outer function: We have f(x) = 3x - 8. This is our main processing machine.
  2. Identify the inner function: We're plugging in g(x) = x + 9 into f(x). This is what we'll be feeding into our machine.
  3. Substitute: Wherever we see an x in f(x), we'll replace it with the entire expression for g(x). So, f(g(x)) becomes 3(x + 9) - 8.
  4. Simplify: Now, let's simplify the expression. Distribute the 3: 3x + 27 - 8. Combine like terms: 3x + 19.

And there you have it! We've found that (f ∘ g)(x) = 3x + 19. This new function takes an input x, multiplies it by 3, and then adds 19. Pretty neat, huh?

But we're not done yet. The next important question is, what's the domain of this composite function?

Determining the Domain of (f ∘ g)(x)

The domain of a function is the set of all possible input values (x-values) for which the function produces a valid output. For simple functions like f(x) = 3x - 8 and g(x) = x + 9, the domain is all real numbers because we can plug in any number and get a real number output. However, composite functions can sometimes have restrictions on their domains.

To find the domain of (f ∘ g)(x), we need to consider two things:

  1. The domain of the inner function, g(x): In this case, g(x) = x + 9 has a domain of all real numbers, so there are no initial restrictions.
  2. The domain of the resulting composite function: Our composite function is (f ∘ g)(x) = 3x + 19. This is a linear function, and linear functions also have a domain of all real numbers. There are no square roots, fractions, or logarithms to worry about, so we can plug in any real number and get a valid output.

Therefore, the domain of (f ∘ g)(x) = 3x + 19 is all real numbers, which we can write in interval notation as (-∞, ∞). Woohoo! We've conquered our first composite function.

b. Finding (g ∘ f)(x)

Now, let's switch things up and find (g ∘ f)(x). This time, we're plugging f(x) into g(x). Remember, the order matters!

  1. Outer function: This time, our main machine is g(x) = x + 9.
  2. Inner function: We're feeding in f(x) = 3x - 8.
  3. Substitute: Replace the x in g(x) with the entire expression for f(x): g(f(x)) becomes (3x - 8) + 9.
  4. Simplify: Combine the constants: 3x + 1.

So, we've found that (g ∘ f)(x) = 3x + 1. Notice that this is different from (f ∘ g)(x), which was 3x + 19. This perfectly illustrates why the order of composition is so important!

Determining the Domain of (g ∘ f)(x)

Let's find the domain of (g ∘ f)(x) = 3x + 1:

  1. Domain of the inner function, f(x): f(x) = 3x - 8 has a domain of all real numbers.
  2. Domain of the composite function: (g ∘ f)(x) = 3x + 1 is another linear function, so its domain is also all real numbers.

Therefore, the domain of (g ∘ f)(x) = 3x + 1 is all real numbers, or (-∞, ∞) in interval notation. Awesome! We're on a roll.

c. Finding (f ∘ f)(x)

Things get interesting when we start composing a function with itself! Let's find (f ∘ f)(x). This means we're plugging f(x) into itself.

  1. Outer function: It's f(x) = 3x - 8.
  2. Inner function: It's also f(x) = 3x - 8!
  3. Substitute: Replace the x in f(x) with the expression for f(x): f(f(x)) becomes 3(3x - 8) - 8.
  4. Simplify: Distribute the 3: 9x - 24 - 8. Combine like terms: 9x - 32.

So, (f ∘ f)(x) = 9x - 32. We've created a new linear function by composing f(x) with itself.

Determining the Domain of (f ∘ f)(x)

Let's figure out the domain:

  1. Domain of the inner function, f(x): We know f(x) = 3x - 8 has a domain of all real numbers.
  2. Domain of the composite function: (f ∘ f)(x) = 9x - 32 is linear, so its domain is also all real numbers.

Thus, the domain of (f ∘ f)(x) = 9x - 32 is all real numbers, which is (-∞, ∞). Great job, guys!

d. Finding (g ∘ g)(x)

Last but not least, let's find (g ∘ g)(x). This is similar to the previous example, but now we're composing g(x) with itself.

  1. Outer function: It's g(x) = x + 9.
  2. Inner function: It's also g(x) = x + 9.
  3. Substitute: Replace the x in g(x) with the expression for g(x): g(g(x)) becomes (x + 9) + 9.
  4. Simplify: Combine the constants: x + 18.

We've found that (g ∘ g)(x) = x + 18. This is another linear function, but it has a different slope and y-intercept than the others we've found.

Determining the Domain of (g ∘ g)(x)

And now, the domain:

  1. Domain of the inner function, g(x): We know g(x) = x + 9 has a domain of all real numbers.
  2. Domain of the composite function: (g ∘ g)(x) = x + 18 is linear, so its domain is also all real numbers.

Therefore, the domain of (g ∘ g)(x) = x + 18 is all real numbers, or (-∞, ∞). We did it! We successfully found all four composite functions and their domains.

Key Takeaways and Why Composite Functions Matter

Wow, we've covered a lot! Let's recap some key takeaways:

  • Composite functions are created by plugging one function into another.
  • The order of composition matters: (f ∘ g)(x) is generally different from (g ∘ f)(x). Remember this, guys!
  • To find the domain of a composite function, consider the domains of both the inner function and the resulting composite function.
  • For these particular linear functions, the domains of the composite functions turned out to be all real numbers, but this isn't always the case. We'll see more complex examples in the future.

So, why are composite functions important? Well, they show up all over the place in mathematics and its applications! They're used to model complex systems, analyze transformations, and even in computer programming. Understanding composite functions is a crucial step in building a strong foundation in math.

I hope this exploration of composite functions with f(x) = 3x - 8 and g(x) = x + 9 has been helpful! Keep practicing, and you'll master these concepts in no time. Keep an eye out for more math adventures coming soon!