# Domain & Range of Composite Functions: Definition & Examples

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• 0:01 A Function
• 1:00 A Composite Function
• 2:17 The Domain
• 3:49 The Range
• 4:41 Lesson Summary

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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Learn what makes a function a composite function and also learn how the parts of a composite function determine its domain. Also in this video lesson, learn about the range of composite functions.

## A Function

To begin, let's first define what a function is. A function produces an output or answer when given an input. Each function performs its own thing to the input. For example, the function f(x) = x - 3 subtracts three from the input to give you an answer; however, the function g(x) = sqrt(x) squares the input to give you an answer.

If we give the same input to both functions, we see that each function produces its own answer. If we give 1 to f(x) = x - 3, we get back a 1 - 3 = -2; but if we give g(x) = sqrt(x) the same 1, we get back a sqrt(1) = 1. Each function produces its own answer. Think of a function as a machine that does work on your input and gives you an output, or answer.

## A Composite Function

Now, what about composite functions? A composite function is the combination of functions. A composite function essentially takes the result of one function and gives it to another. In math, we have a way of writing this. When we are dealing with more than one function, we label each function with a different letter. So one function would be called f(x), and the other g(x). If we are passing the result of g(x) to f(x), we write this composite function as f(g(x)) and we say it as f of g of x. Think of this composite function as one machine's output connected to another machine's input.

We can also write a composite function as one big function. If we are passing the output of g(x) = sqrt(x) to f(x) = x - 3, we can write the composite function f(g(x)) as one big function by plugging in g(x) wherever we see an x in f(x). Our big function will then look like f(g(x)) = sqrt(x) - 3. Do you see how I've replaced the x in f(x) with the sqrt(x), which is g(x)?

## The Domain

Now let's talk about the domain. The domain is your acceptable input. Whatever you give your function has to work, and these values are defined by the domain. Our function f(x) = x - 3 has a domain of all numbers since we can put in any number and get a valid answer.

The function g(x) = sqrt(x), on the other hand, has a restricted domain. This function's domain is all numbers greater than or equal to 0, so only positive numbers. Why is this? If we give this function a negative number, such as -2, we would get an error from this function because the square root of a negative number is not defined.

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