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Compare Properties of Functions Graphically

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  • 0:02 Graphs of Functions
  • 0:36 Linear Functions
  • 4:17 Graphs of Non-Linear Functions
  • 5:54 Vertical Line Test
  • 6:28 Lesson Summary
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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.

Recognizing Graphs of 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.

Fish in an aquarium
several fish in an aquarium with picture of cat

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

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 = 20x, y = ¾x + 3, even the graph of y = 4 is a function.

Linear functions
graphs of linear functions

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:

Linear equation that is not a function
graph of vertical line

This equals this:

This line is not a function, like this cat is not a fish.
imagine of cat and vertical line

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

Let's look at another function:

example of linear function

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

One more function:

graph of function for example

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:

graph of vertical line

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:

These are tables for functions.
tables for functions

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.

Graphs of Non-Linear Functions

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?

graph of parabola

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