Recall that functions are like a black box; they map numbers to other numbers. If y is a function of x, then we write it as y=f(x). And for this function, we have an input, x, and an output, y. So x is our independent variable, and y is our dependent variable. Our input will be anywhere within the domain of the function, and our output will be anywhere within the range of the function. So perhaps it's not too much of a stretch to know that you can combine functions into a big function.
In math, this is known as a composition of functions. Here you start with x, and you use it as input to a function, y=f(x). And you're going to put that as input into a second function, g. So if we have a function y=f(x), and we want to plug it into z=g(y), we can end up with z=g(f(x)). This is a composite function.
When you're looking at composite functions, there are two main points to keep in mind. First, you need to evaluate the function from the inside out. You need to figure out what f(x) is before you figure out what g is. Say we have the function f(x)=3x, and we have another function g(x)= 4 + x. I'm going to find z when x=2. We're going to find f(x) when x=2 for f(2)= 3 * 2, which is 6. Saying g(f(2)) is like saying g(6). We do the same thing and say g(6) = 4 + 6. Well, that's 10, so z is just 10.
The second thing to keep in mind is that g(f(x)) does not equal f(g(x)). There are some cases where it can, but in general, it does not. So if we use f(x)=3x and g(x)=x + 4, then let's look at the case where x=0. Then g(f(0)), where f(0) is 0 * 3 - well that's just zero, so I'm looking at g(0). I plug zero in for x here, and it's just 4. Now, if I look at f(g(0)), that's like saying f(4), and that gives me 12. f(g(0))=12, and g(f(0))=4. Those are not the same. So, g(f(x)) does not equal f(g(x)).
Domain and Range of Composite Functions
What happens to the domain and range of a composite function? Well, if we have the function g(x), we have some domain and some range for g(x). Separately we've got a domain for f(x) and a range for g(x). If I write f(g(x)), then the output of g(x), which is the range, has to be somewhere in the domain of f(x). Otherwise, we could get a number here that f(x) really doesn't know what to do with. What does all this really mean? Consider the function f(x)=sin(x).
The domain of sin(x) is going to be all of x, and the range is going to be between -1 and 1. Now let's look at the function g(x) equals the absolute value of x, or g(x)=abs(x). Again the domain is all of x, and the range is everything greater than 1 or equal to 0. If I take those two - here's my range of sin(x) - what happens to g(f(x))? So g is the absolute value, so I'll have abs(sin(x)).
What's the domain and range of that composite function? If I'm graphing g(f(x)), I'm graphing the absolute value of sin(x), so the graph looks like this. I have a range here that goes from 0 to 1 and a domain that covers all of x. Well, this makes sense. What if I look at f(g(x)), so the function is going to be sine of the absolute value of x, sin(abs(x)).
For the absolute value of x, you can take anything as input, so the domain is going to be all values of x, and the range of abs(x) is going to be zero and up, so anything that's a positive number. Now, sine can take anything, so the range of abs(x) is within the domain of sin(x), but what happens to the output? What is the range of this composite function? Let's graph it - is that unexpected? Now the range is in between -1 and 1, which just so happens to be the range of f(x).
To recap, we know functions map numbers to other numbers, like y=f(x). The domain and range tell us the possible values for the input and output of a function.
Composite functions take the output of one function and use it as input for another function, and we write this f(g(x)). We're going to evaluate f(g(x)) from the inside out, so we're going to evaluate g(x) before we evaluate f(x). And we also know that f(g(x)) does not equal g(f(x)).
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