Let's Learn to compose Functions!

Good Morning class!

I’m Ina, and today we’re going to learn about how to compose functions! I hope you are all very exciting, since this is an important concept, that will return many times as you go into math.

Many people seem to be confused by composing functions, but when you understand the concept behind, it is actually not very hard at all, so bear with me today, and you will be an expert of composing functions!

When composing functions we are always going to have one inner and one outer function. Many times you will see a problem looking similar to the one’s below.

This can look confusing, just because there are multiple parentheses and three variables, but if you break it down, it won’t look as scary, you’ve all seen a function looking just like this one below, right?

It means that when we evaluate the function, we will put in the value of “x” at all places where we can find “x” in the function, for example like this:

See, that wasn’t scary at all! Now when we know that, let’s return to the actual composing. When something looks like the one below, it tells us that we want to evaluate a function, g(z), at the value F(x). In other words, this means that g(z) is our outer function, and F(x) is our inner function. Did I lose you? I’ll explain a little further. Even though the function says g(F(x)) it really tells us that F(x) will be our “x” value in our outer, g(z) function. It means that our input for z, is going to be equal to F(x). You can always think of the outer function being the one on the outside of the parentheses, and the inner function being the one on the inside.

The g(z) function will also have a formula, as you can see below:

When we want to compose the function g(F(x)) we simply mean that whatever value F(x) gives us, will be our input value for z. Since we already evaluated F(x) at x=5, I will use this number in our first composition as seen below:

For me, it’s easier to think of composing as two different evaluations, we want to evaluate our inner function first, in order to see what our outer function is going to be evaluated at. Now, when you’ve again seen that these are skills you already master, we’re going to continue on by doing this algebraically, which can seem hard, but it’s just the same thing again! So, for g(F(x)) we will have:

Isn’t this cool! There’s TWO ways to go about composing a function, we can do it either by:

1) Evaluating our inner function, and then plugging our found value into our outer function

OR

2) Algebraically plug in our inner function as our “z” value for our outer function and then plug in our value.

Whatever way you feel more comfortable with is the way you can use!

So at last today, let’s do one “real world” example:

You are going to paint your house that is 300 square feet. For every square feet, n, you paint, you’re going to need 0.2 gallons of paint. Although, you need to get 5 gallons of paint extra, because you’re really scared you’re going to spill a little while painting and you don’t wanna go back to the store. This relationship can be represented by the function G=g(n)=5+0.2n where G is the total number of gallons of paint you will need and n is the square feet of your house. But you also wonder! How much will all of this cost me? The cost can be represented by C=f(G)=10+10(G/2), meaning that the total cost, C, is equal to 10 dollars which is the base cost for the store to mix the paint, plus 10(G/2), since every half gallon costs 10 dollars. We now want to compose these two functions and solve for the total cost. We do this by composing the two functions.

If you still find this confusing, try going about these steps. Let’s call them Ina’s guide to composing functions!

1) Identify your inner and outer function (i.e which function is inside, and which is outside of the parentheses)

2) Evaluate the inner function at the given value. (You’ve done this a ton of times, so I know you got this!)
3) Take a deep breath, and remember you can do this!

4) Now you feel calm, and you already know what value you are plugging in for your outer function and you can laugh of relief!

5) Plug your value in your outer given function.

6) Solve!

7) Feel accomplished!

Thanks for today! Hope you had as much fun as I did! .