# Composite Function: Definition & Examples

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• 0:03 Composition of Functions
• 1:31 Another Composite…
• 2:51 Composition of Three…
• 3:53 From Composite to…
• 6:00 Lesson Summary

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Lesson Transcript
Instructor: Peter Kosek

Peter has taught Mathematics at the college level and has a master's degree in Mathematics.

One way in which we can combine functions is by forming their composition. A composite function is a larger function that is created by putting one function inside another function.

## Composition of Functions

Let's suppose we have two functions - say f(x) = x^2 and g(x) = x + 3. We can form a new function, called their composition, by putting one function inside the other. Let's see what happens when we trying putting g(x) inside f(x).

Instead of plugging in an x into the function f(x), we'll plug in g(x) ,and we'll write it as f(g(x)). When you're composing functions, you should always remember to work from the inside out. Since we know that g(x) = x + 3, we can substitute that in. Therefore, f(g(x)) = f(x + 3). To finish our composition, we use the fact that f(x) = x^2 to evaluate f(x + 3) = (x + 3)^2. Now, we're done! Therefore, f(g(x)) = f(x + 3) = (x + 3)^2.

Now, you might be thinking, ''How did you evaluate f(x + 3) = (x + 3)^2?'' Here's a nifty trick I use to correctly do this. First, I write f'(x) = x^2. Then, everywhere I see an x, I write a set of empty parenthesis. So I would write, f( ) = ( )^2. I now see that whatever I have replacing the x on the inside of the f(x), I want to write that everywhere I have empty parenthesis. That's how we get f(x + 3) = (x + 3)^2.

## Another Composite Function Example

Let's look at another composite function example.

Suppose f(x) = x - 5 and g(x) = x^2 + x. Let's find g(f(x)).

Remember, we have to work inside out! Let's first substitute for f(x). We get g(f(x)) = g(x - 5).

Now, if you like my trick of writing the empty parenthesis, we can do that and get g( ) = ( )^2 + ( ). Plugging in the x - 5, we get g(x - 5) = (x - 5)^2 + x - 5. If we would like to simplify this expression, we would get the following: (x - 5)^2 + x - 5 = (x - 5)(x - 5) + x - 5 = x^2 - 5x - 5x + 25 + x - 5 = x^2 - 9x + 20.

Therefore, g(f(x)) = g(x - 5) = x^2 - 9x + 20.

We could also find g(f(x)). For this one, we see, f(g(x)) = f(x^2 + x) = x^2 + x - 5. Notice that g(f(x)) ≠ f(g(x)). Therefore, it's important that we compose our functions in the correct order!

## Composition of Three or More Functions

So far, we've been looking at the composition of two functions. Does it only work with two functions? Can we compose three functions? What about four functions?? What about 3,456,193 functions?

We can compose as many functions as we'd like. Let's look at an example.

Suppose f(x) = 2x + 1, g(x) = x^2 + x, h(x) = 3x. Find f(g(h(x))).

Remember, when we evaluate the composition of functions, we have to work inside out. Therefore, f(g(h(x))) = f(g(3x)) = f((3x)^2 + 3x) = 2((3x)^2 + 3x) + 1. Simplifying this expression, we get:

2((3x)^2 + 3x) + 1 = 2(9x^2 + 3x) + 1 = 18x^2 + 6x + 1. Therefore, f(g(h(x))) = 18x^2 + 6x + 1.

## From Composite to Separate Functions

What would you do if someone was to give you a composite function and ask you to work backwards? That is, if you were given a composite function, could you determine what functions were initially composed with one another to create the given function?

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