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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

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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.

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.

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)*?

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.

Because we have more than one function in a composite function, we have to consider the domain of each function when we consider the domain of the composite function. For our composite function *f(g(x))* = sqrt(*x*) - 3, even though the domain of our function *f(x)* = *x* - 3 is all numbers, we are still restricted because we also have to consider the domain of *g(x)* = sqrt(*x*). Because the domain of *g(x)* = sqrt(*x*) is restricted to only positive numbers, we have to include this in our domain. So our domain for the composite function becomes only positive numbers. We have to include all the restrictions from each function for the domain of our composite function.

The **range** of a function is the possible outputs. We specify the range for functions that produce outcomes that are limited in some way. For our composite function *f(g(x))* = sqrt(*x*) - 3, the range is all numbers greater than -3. How did we get all numbers greater than -3? We got this by looking at the components of our function.

First, the sqrt(*x*) tells us that our output can only be numbers greater than 0. But then we have the -3 following this. The -3 tells us that whatever answer we get from the square root is lowered by 3, so our lowest possible answer is -3. Because that is our lowest possible answer, our range is all numbers greater than this lowest possible number, -3.

What have we learned? We've learned that where a **function** produces an output, or answer, when given an input, the **composite function** is the combination of functions. If you liken a function to a machine that receives some input and spits out the output, then a composite function can be likened to a series of machines where the output of one machine is connected to the input of the next machine.

In math, we write composite functions using a different letter for each separate function. So if our two functions are *f(x)* = *x* - 3 and *g(x)* = sqrt(*x*), then the composite function where the output of *g(x)* is the input for *f(x)* is written as *f(g(x))*.

The **domain**, acceptable inputs, is determined by looking at what is an acceptable input for each of the functions and then including all the restrictions. The **range**, possible outputs, is determined by looking at the composite function and seeing if it has a minimum or maximum answer or an answer it cannot be.

Allow this video lesson's content to help you to:

- Define and exemplify a function and a composite function
- Understand the relationship between a function and its domain and range

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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

- How to Add, Subtract, Multiply and Divide Functions 6:43
- How to Compose Functions 6:52
- Applying Function Operations Practice Problems 5:17
- Compounding Functions and Graphing Functions of Functions 7:47
- Domain & Range of Composite Functions: Definition & Examples 5:58
- Understanding and Graphing the Inverse Function 7:31
- One-to-One Functions: Definitions and Examples 4:11
- How to Determine the Limits of Functions 5:15
- One-Sided Limits and Continuity 4:33
- Function Application for the Real World 5:36
- Using Quadratic Formulas in Real Life Situations 5:22
- Go to Function Operations

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