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Common Core Math Grade 8 - Functions: Standards5 chapters | 19 lessons

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Lesson Transcript

Instructor:
*John Sepanski*

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

In this lesson, we will cover the characteristics of graphs of functions. We will employ the vertical line test and understand why some graphs do not qualify as functions.

A lot of people in the world have a pet. Some people have dogs, some have cats, some have birds, some have fish and some even have tarantulas! But for this lesson, let's just talk about fish. A visit to a pet store will amaze you with how many different types of fish there are to own. There are Bettas, angelfish and catfish.

Catfish? Catfish don't look like that! That's not a fish at all. That's a cat in an aquarium!

Functions are like that. Pictures of functions, or graphs of functions, have specific characteristics that make it obvious that they are functions.

Linear functions are really easy to recognize. As a matter of fact, almost every graph of a line is a function. As a matter of fact, all of these are functions: *y* = 20*x*, *y* = ¾*x* + 3, even the graph of *y* = 4 is a function.

But there's one type linear equation that is not a function. Can you think of what it might be? It would have to be something different, like a cat in an aquarium!

Take a look at this graph:

This equals this:

Why? Do you remember the definition of function? A **function** is an algebraic relationship where every input has one distinct output. Remember that the input had a special name: **Domain** is the set of input values (*x*), and **range** is the set of output values (*y*). So the definition of function means that for every *x* value there is exactly one *y* value, no more or no less.

This is kind of like describing what a pet fish is. A fish has eyes, a mouth, gills and fins. In algebra, it works very much the same.

*y* = 2*x* - 2. For *x* = 0, *y* = -2. For *x* = 1, *y* = 0, and for *x* = 2, *y* = 2.

Let's look at another function:

For *x* = 0, *y* = 3. For *x* = 4, *y* = 6. For *x* = 8, *y* = 9.

One more function:

For *x* = 0, *y* = 4. For *x* = 1, *y* = 4. For *x* = 2, *y* = 4. *y* is always going to equal 4! No matter what we put into our function machine, it always produces a 4.

Does this satisfy our definition of a function? Function is an algebraic relationship where every input has one distinct output. When *x* = 0, *y* = 4. 4 is the only output for *x* = 0. There is no other. When *x* = 1, *y* = 4. 4 is the only output for *x* = 1. There is no other. When *x* = 2, *y* = 4. 4 is the only output for *x* = 2. There is no other. Every time we input something, we're only going to get one answer.

Let's take a look at a relationship that is not a function:

When *x* = 1... How can we put that into the equation? We can't put it in there. *x* is always 4. Okay, let's start over. When *x* = 4, what does *y* equal? Look at the graph. Let's just pick one. *y* = 1. When *x* = 4, then *y* = 2. When *x* = 4, then *y* = 10.

Look at the t-table (on the right side of the image above). Now let's look at the table for a function below. Check out another, and another, and another:

Do you notice the difference?

Function is an algebraic relationship where every input has one distinct output. Notice that on all of the functions, when we put in 1 input we get a different output? A distinct output? But when the relationship is not a function, every time we put in a different value of *y*, we get the same value for *x*. Every element of the range has the same value in the domain.

Let's look at some graphs of non-linear functions. Well, this is obviously a graph, but is it a function or not a function?

Let's look a little closer. When *x* = -2, *y* = 4. When *x* = -1, *y* = 1. When *x* = 0, then *y* = 0. When *x* = 1, *y* = 1. When *x* = 2, *y* = 4 Do the repeating values of the range, *y* = 1 and *y* = 4, cause this not to be a function?

Function is an algebraic relationship where every input has one distinct output. Here's the question: If I input -1 into my function, will I ever get anything other than 1? If I input 0 into the function, will I ever get anything other than 0? If I input 2 into the equation, will I ever get anything besides 4? That is the reason this equation is a function!

Let's look at one more non-linear equation. Here's the equation for *y*^2 = *x*. Let's look to see if this is a function. When *x* = 0, then *y* = 0. So far, so good! When *x* = 1, what does *y* equal? Look at the graph:

It indicates that when *x* is 1, *y* is both 1 and -1. This is not a function! A function is an algebraic relationship where every input has one distinct output. This relation shows that for *x* = 1 there are two possible outcomes. Therefore, it is not a function.

There is a very easy way to determine if a graph is a function. We call it the vertical line test. If we have a graph of a relationship and slide the vertical line across it, it will only hit one point at a time if the relation is a function. If the vertical line ever touches more than one point on the graph at a time, the relationship is not a function.

How about this one?

If you think about it, this relation has an infinite number of outputs for this 1 input! This is the ultimate non-function!

A **function** is an algebraic relationship where every input has one distinct output. This produces a unique graph. The graphs of functions may be linear. Each value of *x* indicates only 1 value of *y*.

There is one type of linear relationship that is not a function, and that is when the graph makes a vertical line. This graph shows that for *x* = 4, there are multiple values of *y*:

The easiest way to check to see if a graph represents a function is to use the vertical line test. If the vertical line touches only one point on the line at a time, then you know you have a function. If the vertical line ever touches more than one point at a time, the graph is not a representation of a function.

Now recognizing functions is as easy as telling the difference between a fish and something that is not a fish!

After this lesson, you'll be able to:

- Define function
- Explain how to identify whether a graph is a function

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Common Core Math Grade 8 - Functions: Standards5 chapters | 19 lessons

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