Evaluating composite functions: using tables | Mathematics III | High School Math | Khan Academy

• [Voiceover] So we have some tables here that give us what the functions f and g are when you give it certain inputs. So, when you input negative four, f of negative four is 29. That's going to be the output of that function. So we have that for both f and g, and what I want to do is evaluate two composite functions. I want to evaluate f of g of zero, and I want to evaluate g of f of zero.

So like always, pause the video and see if you can figure it out. Let's first think about f of g of zero. F of g of zero. What is this all about? Actually let me use multiple colors here. F of g of zero. Well, this means that we're going to evaluate g at zero, so we're gonna input zero into g. Do it in that. So we're gonna input zero into our function g, and we're going to output, whatever we output is going to be g of zero. I'll write it right over here, and then we're going to input that into our function f. We're going to input that into our function f, and whatever I output then is going to be f of g of zero. F of g of zero. F of g of zero. I wrote these small here so we have space for the actual values.

So first let's just evaluate, and if you are now inspired, pause the video again and see if you can solve it. Although, if you solved it the first time, you don't have to do that now. What's g of zero? Well, when we input x equals zero, we get g of zero is equal to five. So g of zero is five. So that is five. So we're now going to input five into our function f. We're essentially going to evaluate f of five. So when you input five into our function. I'm gonna do it in this brown color. When you input x equals five into f, you get the function f of five is equal to 11. So this is going to be 11.

So, f of g of zero is equal to 11. Now, let's do g of f of zero. So now let's evaluate. I'll do this is different colors. G, maybe I'll use those same two colors actually. So now we're going to evaluate g of f of zero. G of f of zero, and the key realization is you wanna go within the parenthesis. Evaluate that first so then you can evaluate the function that's kind of on the outside.

So here we're going to take zero as an input into the function f, and then whatever that is, that f of zero, we're going to input into our function g. We're going to input into our function g, and what we're going to be, and then the output of that is going to be g of f of zero. So, let's see, what is f of zero? You see over here when our input is zero, this table tells us that f of zero is equal to one. So f of zero is equal to one. F of zero is equal to one.

So now we use one as an input into g. We're now evaluating g of one, or I can just write this. This is the same thing as g of one. G of one. Once again, why was that? 'Cause f of zero is equal to, f of zero is equal to one. And let me, I wrote those parenthesis too far away from the g. This is the same thing as g of one. Because once again f of zero is one.

Now what is g of one? Well, when I input one into our function g, I get g of one is equal to eight. So this is going to be equal, this is equal to eight, and we're done. And notice these are different values, because these are different composite functions. F of g of zero is 11, and g of f of zero is eight.