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Composing Functions: Definition and Examples

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Instructor: John Sepanski

John has taught 6th Grade Mathematics through Geometry and has a Master's degree in Education

A function represents the relationship between different variables and it is used to represent some examples of real-world phenomena. Learn the definition of a function, how to compose functions, the domain, range, and notation of a function, and how to use a function table. Updated: 10/09/2021

Composing Functions

Have you ever used emoticons?

You know, emoticons are facial expressions that you can use when you text or type on some apps.

For instance, if you type, :), you will get


Smile emoticon


Or if you type ;), you will get


Wink emoticon


Some people get so used to using emoticons that it's just something that they naturally type.

As a matter of fact, it could be said that


Smile emoticon


Is the outcome of typing :).

An input of ;), will produce an output of


Wink emoticon


Mathematical functions are exactly like that.

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  • 0:06 Composing Functions
  • 0:50 Function Definition
  • 2:04 Domain, Range and…
  • 3:15 Function Table
  • 5:12 Lesson Summary
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Function Definition

A function is a mathematical relationship where there is exactly one output for each input.

The set of the input values is called the domain and the set of output values is called the range.

Maybe this is going a little too fast, and you're beginning to feel a little o.O


Confused emoticon


Don't worry because I'm going to make it so easy that you're going to <3 math!

Let's examine the function definition more closely.

A function is a mathematical relationship where there is exactly one output for each input. Think about the word, 'relationship.' Maybe some of you are in a relationship. A relationship is literally a state or status of being connected. In mathematics, this state of being connected is often expressed through the use of an equals sign.

So that must mean that an equals sign is just a mathematician's way of typing <3.

Back to the definition.

A function is a mathematical relationship where there is exactly one output for each input.

Input. That's like the text you type into your device to produce an emoticon. The emoticon is the output.

Domain, Range and Function Notation

In math, the input and the output are identified with special names: domain and range. Remember, the domain is the set of all input values, and the range is the set of all output values.

Let's see how these fit into a mathematical expression.

:v


pacman emoticon


To display functions, we use a system of mathematical symbols known as functional notation. When we write in functional notation, it's kind of like writing in text. We use specific expressions to mean certain things. For example, we write functions like this:

f(x) = 2x + 2

This is read, f of x equals 2x + 2.

So, f of x is the range.

x itself is the domain.

In other words, if we insert the domain into a function, as a result of the function, we get the range.

Sometimes the range is indicated with the variable y.

So, sometimes a function like f(x) = 3x + 4 is written as y = 3x + 4.

Function Tables

A math function table can be used to plot the possible outcomes of a function. Function tables are sometimes called T-Tables.


function table 1


This particular table is for the function:

y = 3x + 4

If you input 0 into the function in the place of x, the output (or y value) is 4. If you input 1 into the function, the output is 7. If you input 2 into the function, the output is 10. And if you input 3 into the function, the output is 13.

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