relations and functions

relations and functions

An ordered pair consists of two numbers that tell us where on a graph the point is.

An ordered pair is written with the X value first, followed by the Y value second. The X value are the numbers on the X-axis or left to right. The Y value are the numbers on the Y-axis or up and down. An ordered pair is written as (X,Y).

What is a relation?

Just like IRL(in real life), we have relationships with friends and family. If there is more than one ordered pair(X,Y), they have a relation. Typically these ordered pairs would be on the same coordinate plane or graph, so that we can see what relation they have to each other. Two examples of the relations they might have would be the domain and range. Domain and range are helpful to know what area of the graph you will be looking at.

What are domain and range?

I'm glad you asked. Domain represents the input values or the X values. So as we look from left to right on our graph, the first X value is the start of our domain and the last X value is the end of our domain. Range represents the output values or the Y values. So as we look up and down on our graph, the bottom Y value is the start of our range and the top Y value is the end of our range.

Notice how the domain and range correspond to these relations.

The domain is the X values of our ordered pairs listed in numerical order. The range is the Y values of our order pairs listed in numerical order. You don't have to repeat any numbers in the domain or range. If you wrote it once then that is good enough. Try some...

Multiple Choice

What is the domain of these relations?

(3,4) (2,1) (-4,-8) (4, 7)

• 4, 1, -8, 7

• -4, 2, 3, 4

• 1, 2, 3, 4

• -8, 1, 4, 7

Multiple Choice

What is the range of these relations?

(-1, -7) (1, 7) (2, -2) (-7, 1)

• -1, 1, 2, -7

• -7, -1, 1, 2

• -7, -2, 1, 7

• -1, 1, -7, 7

Input is our X value

Output is our Y value.

When you have an ordered pair like (3,7) the X value represents the input and the Y value represents the output. Think of this like a vending machine. You input your money and you get(output) your snack or drink. Which item you get depends on what buttons you push. Function inputs are like what buttons we pushed. I push button 3 and i get snack 7.

Functions will have one unique output for each input.

In the vending machine, if you input your money, you can only pick one snack or drink, and what you pick is what you get. If you get something else, then the machine is broken. In math a broken machine is not a function. Anytime you input an X value, you should get the same Y value or it is not a function.

Notice the difference between a function and not a function.

The left side of the picture shows functions. You input one value and you get a unique value as an output. Like vending machines that have the same snack in two different spots, so can you get the same output for two different inputs. But if you input something and get a different snack each time, then that is wrong and in math that is not a function.

Is this a function?

The last pictures had mapping arrows, but this relation does not. Remember the X is the input and the Y is the output; (X,Y). What you need to look at is the X values first. Are there any X values that are the same? If no, then it is a function. If yes, then do the Y values match, as in, is it the same ordered pair values? if yes then it is a function, if not then it is not a function. Notice X=-2 for two of the ordered pairs, but the Y values are different.

Multiple Choice

Is this relation a function?

• no

• yes

• why so serious

• maybe

Multiple Choice

Is this relation a function?

• yes

• no

• you tell me

• maybe

Multiple Choice

Is this relation a function?

• no

• yes

• so many numbers

• maybe

Multiple Choice

is this relation a function?

• yes

• no

• maybe

• are we done yet?

What if we have a picture of a graph, How do we tell if it is a function?

The same rules apply, each X value can have only one Y value. All these graphs are functions. How would you know just looking at them? There is a simple test we can use visually. The vertical line test. A vertical line is a line that goes straight up and down.

Notice the blue line in these graphs.

The blue line is a vertical line. The red is the graph of an equation. The dots represent where on the graph the vertical line test touches our equation. Notice the vertical line crosses the X-axis at only one spot, that represents what X value we picked, so there should only be one Y value. The left picture has two dots and two different Y values. It is not a function. The right has one dot, so it is a function.

You can use a pencil or ruler to help visualize the vertical line test.

Notice the top graph, a ruler held vertically, only touches our blue equation once at any spot, so it is a function. Notice the the bottom graph, a ruler held vertically, touches three spots at once on our blue equation, so it is not a function. Try some...

Multiple Choice

Is this a function?

• yes

• no

• maybe

• nice picture

Multiple Choice

is this a function?

• no

• maybe

• yes

• is that a 2 or an o?

Multiple Choice

Is this a function?

• maybe

• yes

• no

• is this a S or a 2?

Multiple Choice

is this a function?

• this is definitely a M

• yes

• no

• maybe