Hi, and welcome. My name is Anthony Varela. And today we're going to talk about the function of a function.
So we can also call these composite functions. And this is where we have more than one function such that a function is the input of another function. And we'll also take a look at composite functions on a graph.
So a composite function is the combination of functions such that the output of one function is the input of another. So for example, if we have two functions, f and g, we can create a composite function by putting one function inside of another. So here we see that g of x is the input or arguments to the function f.
Now we can also write this using an open circle. And be very careful to actually make this an open circle. If you fill it in or make it look like a filled in dots, that could represent function multiplication. And that's very different from a composite function.
But in both cases, this reads f of g of x. So those are a couple of different ways to write out a composite function in function notation.
So how do we evaluate a composite function? So given that f of x equals negative 2x plus 3, and g of x equals x squared, I'd like to evaluate f of g of 2. So how I normally would evaluate a function is I would take a look at the outside function f. And then I would just substitute the value that I see in my argument in for its expression.
Well, I see another function in here. So we actually need to evaluate g of 2 first. And once I have a value of g of 2, I can substitute that in for x and use that in f of x.
So let's evaluate g of 2. Well, g of 2 is 2 squared-- just substituting 2 in for x. 2 squared is 4. So f of g of 2 is the same as f of 4.
And so now we'll take a look at our function f and replace our x's with 4. So f of 4 equals negative 2 times 4 plus 3. Well, that's negative 8 plus 3, or negative 5. So f of 4 equals negative 5. That means that f of g of 2 is also negative 5.
Now another thing that you could do is you can do the substitution at the very end. And you can take are expression for g of x and put that into the argument of f. So this would be f of x squared, because g of x and x squared are the same thing.
So we would be taking our function f, and instead of writing x, we'd write x squared. So this would be negative 2 times x squared plus 3. Well, now we can go ahead then and substitute in our x with 2. So we would have negative 2 times 2 squared plus 3.
2 squared is 4, times negative 2 is negative 8. Adding 3, we get negative 5. So the same answer in both cases.
Next, we're going to use a graph to evaluate the function of a function at a certain point. So here I have the graph y equals f of x. And I'd like to find f of f of 2. So how can I do this on the graph?
Well, first I'm going to find f of 2. So find f of a, whatever value you're given. But then once I have that value, I'm going to use that as the input to the function again. So I'm evaluating this function twice.
So first, I need to find f of 2. So to find f of 2, I'm going to locate two on the x-axis and go out to my function. So f of 2 equals 1.
So now I'm going to use 1 as the input to the function again. So I'm going to find f of 1. So that means locating x equals 1, and then going to my function. And that is negative 2. So because f of 1 is negative 2 and because f of 2 is 1, I know then that f of f of 2 is negative 2.
Lastly, we're going to practice writing some composite functions. So here we have f of x equals x minus 3, and g of x equals negative 2x squared. And we're going to write two composite functions, one where we have f of g of x. And then we have g of f of x.
So I'm swapping the functions around. And we're going to see how this affects how we write our composite function. Well, to interpret f of g of x, I could say this is f of negative 2x squared because that's what g of x is.
So now I'm going to look at f of x. And instead of writing x, I'm going to write negative 2x squared. So that would then be negative 2x squared minus 3.
And I can write that the same without the parentheses because there's no distribution or anything like that. So f of g of x equals negative 2x squared minus 3.
Well, now let's write g of f of x. So to interpret g of f of x, I'm just going to replace f of x with x minus 3. That's the argument of our function g.
So now, instead of writing negative 2x squared, I'm going to write an x minus 3 when I see x. So g of x minus 3 then would be negative 2. And I have x minus 3 instead of x. And that entire quantity is squared. So be sure to group this in parentheses whenever we have that exponent there.
Well, now I'm going to square x minus 3. So x minus 3 times x minus 3 is x squared minus 6x plus 9. That's just using FOIL to multiply that.
All of this is being multiplied by negative 2. So I'm going to double every term and then change their sign. So I have negative 2x squared plus 12x minus 18.
So notice that order matters when you're writing composite functions. Negative 2x squared minus 3 is not the same as negative 2x squared plus 12x minus 18. So order matters when writing composite functions.
So let's review the function of a function. Well, we can also call this a composite function. It's where we have more than one function and we have a function as the input to another function. And here's another way we can write this-- we read it f of g of x.
Using a graph to evaluate the function of a function, first, you can find f of a, and then use that as the input to the function again. And remember, when you're writing composite functions, f of g of x is not the same as g of f of x.
Thanks for watching this tutorial on the function of a function. Hope to see you next time.
