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Function of a Function

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Today we're going to talk about functions of a function, or composite functions. So we're going to start by talking about what a composite function is, and then we'll do some examples evaluating and writing composite functions. So let's start by talking about what a composite function is. I've got two functions-- g of x, which is equal to x minus 3, and f of x, which is equal to 5x squared. If I want to find a composite function of these two functions, I'm looking for the function of a function. So the output of one function becomes the input of another function.

So for example, we could find the composite function f of g of x. The way that I would say this is f of g of x. And f of g of x is going to be equal to 5 x minus 3 squared. And the way that I found that it's 5 times x minus 3 squared is I have my first function, g of x, which I know is x minus 3. That output becomes the input of my second function, 5x squared. So that replaces my x in my function f of x. So then I have 5 times x minus 3 squared.

Another way of notating f of g of x would be f, a small circle, g, and then x in parentheses. So these are both ways of notating f of g of x. So let's do an example evaluating composite functions. I've got the function f of x, which is equal to 8 x squared minus 4, and the function g of x, which is equal to x minus 1, x plus 1. I want to find f of g of 3. To do that, I'm going to start by evaluating the innermost function, g of x, and I'm evaluating it when x is equal to 3.

And the output, whatever g of 3 is equal to, is going to be my input from my f function. So that's what I'll substitute in for x in my function f of x. So I'm going to start by finding g of 3, which means I'm going to be replacing 3 for my x variable. So this would become 3 plus 1, or 4. So g of 3 is equal to 4.

So now 4 it's going to become my input for my function f of x. So I'm going to input 4 for x in my function f of x. So this will becoming 8 times 4 squared minus 4. Simplifying, 4 squared is 16, 8 times 16 is 128, and 128 minus 4 will give me 124. So I found that f of 4 is 124, which is the value of f of g of 3.

So let's see how we can evaluate a composite function on a graph. I've got a graph of the function f of x, which is 4x minus 4. So let's say I want to evaluate the function f of f of x for x is equal to 1. So I'm going to start with my input of 1, evaluate f of 1, and then use that output to be the input of a second function, f of x.

So if x is equal to 1 on my graph, I'll go into 1 on my x-axis. And to find the value of f of x, I'll go over to my y-axis, which is also f of x, and I see that that is 0. So I know that f of 1 is equal to 0. Now if I want to find f of f of 1, since I know f of 1 is 0, I'm finding f of 0. So again, I'll go to my graph. My input now is 0. So I'll go to 0 on my x-axis, and go to my graph. And I find that my output value, or my value for my value for f of x, is negative 4. So f of 0 is negative 4.

So I've found that the composite function f of f of 1 is equal to negative 4, So let's do an example writing a composite function. I've got the function f of x, which is equal to x squared plus 4x. And the function g of x, which is equal to x plus 3. If I want to find f of g of x, I'm going to start by writing my function g of x as the input for my function f of x.

So I'm going to substitute my expression for g of x into my variable for x, as this is now going to become the input for my f function, my function f of x. So this is going to be equal to x plus 3 squared plus 4 times x plus 3. I just substituted my expression, x plus 3, in for my x variable in my function f of x.

So now I can simplify. x plus 3 squared, I know means x plus 3 times x plus 3. And then I have plus 4 times x plus 3. I'm going to multiply this using FOIL. And then I'm going to distribute my 4 to both terms on the inside. OK now I'm going to simplify by combining like terms.

So I have a 3x, a 3x, and a 4x. That will give me 10x. I have an x squared and nothing else to combine it with, so I'll just bring that down. And then I have a 9 and a 12, so I'll add those together to give me 21. So I found that f of g of x is equal to x squared plus 10x plus 21.

So now let's do another example with composite functions. My function f of x is still x squared plus 4x, and g of x is still x plus 3. But now I want to find g of f of x. So now f of x will become the input for my function g of x. So I'm going to take my expression for f of x, and I'm going to input that into my expression for g of x.

So instead of x, here I'll have x squared plus 4x, and then plus 3. So to simplify this, I don't have any terms that I can combine. So I've found that g of f of x is equal to x squared plus 4x plus 3.

Now if I remember, what we got for our previous answer in the previous example, f of g of x, we found was equal to x squared plus 10x plus 21. So we can see that when you're finding composite functions, the order does matter. g of f of x is not equal to f of g of x.

So let's go over our key points from today. A composite function is a function of a function. We used the notation f of g of x, or f of g of x. In a composite function, the output of the innermost function becomes the input of the outermost function. And it is not always true that f of g of x is equal to g of f of x.

So I hope that these key points and examples helped you understand a little bit more about function of a function. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.