*John Sepanski*Show bio

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

Lesson Transcript

Instructor:
*John Sepanski*
Show bio

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

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

Or if you type ;), you will get

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

Is the outcome of typing :).

An input of ;), will produce an output of

Mathematical functions are exactly like that.

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

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.

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

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*) = 2*x* + 2

This is read, *f* of *x* equals 2*x* + 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*) = 3*x* + 4 is written as *y* = 3*x* + 4.

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

This particular table is for the function:

*y* = 3*x* + 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.

As you can see, there is a direct mathematical relationship between the input and the output. There is only one output for every input.

This function table is kind of like this other one.

As we enter :), we expect to get:

If we enter ;), we expect to get:

If we enter the domain of :( we expect the output of:

and if we enter the domain of <3, we get the range of:

One last point: each time we indicate an emoticon by typing :) we expect to get:

We never expect to get:

What that means is that :) never equals:

Mathematical functions are exactly like that. The function always produces the expected output for the specific input.

So, when we enter 0 in place of x into the function *y* = 3*x* + 4, we will always get 4 as the *y* value. If we enter 0 into our function again, we will never get anything other than 4.

Nothing else would ever be expected.

Let's review. 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**.

To display functions, we use a system of mathematical symbols known as **functional notation**. For example:

*f*(*x*) = -2*x* + 4.

This is read as *'f* of *x*' equals -2*x* plus 4.

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

These tables look like this:

As we enter zero into the function, we get an output of 4. When we enter 1 into the domain, we get an output of 2, and when we have a domain of 2 into the function, we get a range value of zero.

When you began the lesson, you may have been feeling a little o.O

But as we worked through it, you became more and more :3

Hopefully, by now, you're just cool 8/ with functions!

As this lesson progresses, develop your capacity to:

- Recognize a function
- Define domain and range in terms of functions
- Demonstrate proper function notation
- Use a T-Table when working with functions

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