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General Studies Math: Help & Review8 chapters | 85 lessons

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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.

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.

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 - 5*x* - 5*x* + 25 + *x* - 5 = *x*^2 - 9*x* + 20.

Therefore, g(f(*x*)) = g(*x* - 5) = *x*^2 - 9*x* + 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*)) â‰ f(g(*x*)). Therefore, it's important that we compose our functions in the correct order!

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*) = 2*x* + 1, g(*x*) = *x*^2 + *x*, h(*x*) = 3*x*. 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(3*x*)) = f((3*x*)^2 + 3*x*) = 2((3*x*)^2 + 3*x*) + 1. Simplifying this expression, we get:

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

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?

Suppose somebody gave you the function h(*x*) = (3*x* - 5)^2 and asked you to find two functions - f(*x*), g(*x*) - such that h(*x*) = f(g(*x*)).

In order to decompose a composite function, it helps to first recognize a possible inner function that may have been the function we plugged into the other. In this case, our inner function will be g(*x*), since that one is in the tummy of f(*x*). Looking at h(*x*) = (3*x* - 5)^2, we decide that 3*x* - 5 could be our inner function. Great!

Now, let's choose our outer function, which in this case is f(*x*). To find f(*x*), simply look at h(*x*) = (3*x* - 5)^2, cover up the inner function everywhere you see the inner function with your hand, then write down what you see and put an *x* everywhere your hand is. By successfully doing this process, we should find that f(*x*) = *x*^2.

The best part about decomposing a composite function is that we can always check our answer to see if it's correct. Let's check it! We see f(g(*x*)) = f(3*x* - 5) = (3*x* - 5)^2 = h(*x*). Fantastic! Our choices were correct.

What if your friend was to say, ''I chose my inner function to be g(*x*) = 3*x*''? Is your friend off to a wrong start and about to crash and burn? Not at all!

If you choose g(*x*) = 3*x*, then we can repeat the same process as before and cover up the 3*x* with our hand and rewrite the remaining pieces of the function while writing an *x* where our hand is. We would then find f(*x*) = (*x* - 5)^2. We can check this answer and see that f(g(*x*)) = f(3*x*) = (3*x* - 5)^2 = h(*x*), which is exactly what we wanted.

Therefore, when decomposing composite functions, you may have multiple correct answers.

Remember, when you are asked to find the composition of functions, work from the inside out. Take it one step at a time and you'll be fine.

When you're given a composite function and asked to work backwards:

- First, find an inner function.
- Then, use the hand cover up trick to properly find the outer function.
- Finally, rejoice! You did it!

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