Relations and Functions

Any set of points in a table or on a graph is called a relation. Some relations are functions. In this lab you will learn the difference between relations and functions. You will also learn about the domain and range of relations and functions.

Relations

Question 1. The set of inputs ($x$-values) of a relation is called the domain of the relation. By looking at the table to the right, you can see that the domain of the relation on the grid to the left is the set $\{-1,1,2,3,4\}$. The set of the outputs ($y$-values) of a relation is called the range of the relation. What is the range of this relation?

Range: { , , , , }
$x$ $y$
-11
1-1
23
3-2
42

Click on to see a graph of the next relation. Find the domain and range of this relation using the table to the right. Remember that you don't need to repeat numbers in a set.

Domain: { , , , , }
Range: { , , , , }
$x$ $y$
-20
-11
-1-1
02
0-2
11
1-1
20

Click on and find the domain and range of this relation.

Domain: { , , , , }
Range: { }
$x$ $y$
-22
-12
02
12
22

Question 2. Sometimes instead of a table you have to look at a graph to find the domain and range of a relation. Click on and look at the grid to the left. This time the graph is not just a few points, but the collection of points that make the red curve.

Click inside the grid and drag the vertical bar. You see that the curve goes from $x=-2$ to $x=4$. So the domain of this relation is $\{x:-2≤x≤4\}$. What is the range of this relation?

Range: $\{y:$ $≤y≤$ $\}$

Click on and find the domain and range of this new relation using its graph.

Domain: $\{x:$ $≤x≤$ $\}$
Range: $\{y:$ $≤y≤$ $\}$

Question 3. Click on and look at the graph of $y=x^2$. It seems that the curve goes from $x=-3$ to $x=3$. Click inside the grid and drag the vertical bar to $x=4$. To the right of the grid you see that $y=16$, so even though we can't see it on this grid, there is a point on the graph with $x$-coordinate 4.

Drag the vertical bar to $x=-5$. Is there a point on the graph with $x$-coordinate −5?

Do you think that there is a point on the graph with $x$-coordinate 40?

Click on to look at more of the graph. Is your answer to the previous question correct?

Is there a point on the graph with $x=-25,000$?

Is there a limit to how large or small the $x$-value of a point can be on the graph of $y=x^2$?

When there is no limit on what $x$ can be, we say that the domain of a relation is “all real numbers”. What is the domain of $y=x^2$?

What is the smallest $y$-coordinate of any point on this curve?

What is the range of this relation?

Range: $\{y:$ $≥$ $\}$

Functions

A relation is called a function if there is exactly one output ($y$-value) for each input ($x$-value). If a relation has two or more outputs for a single input, it is not a function. For the two relations shown below, the one on the right isn't a function because there are two outputs for each input greater than 0. The relation on the left is a function.

A Function Not a Function
A FunctionNot a Function

Question 4. Click on . Click inside the grid then drag the vertical bar and place it on the points on the graph. What are the coordinates of these points? &

What is the input ($x$-value) for each of the two points in the previous question?

How many outputs ($y$-values) are there for the input ($x$-value) you found in the previous question?

Is the relation shown to the left a function?

What is the domain of this relation?

Domain: $\{x:$ $≤$ $\}$

Click on . Use the vertical bar in the grid to decide if the relation is a function. Is it a function?

The domain of this relation is all real numbers. What is its range?

Range: $\{y:$ $≤y≤$ $\}$

Click on . Use the vertical bar in the grid to decide if the relation is a function and to help find its domain and range.

Function?
Domain:
Range:

Click on . Is this relation a function?

Question 5. You can also decide if a relation is a function by looking at a table. In the table to the right the input −2 has two outputs (1 and −1). You can see this on the graph by clicking . There are two points that have $x$-coordinate −2.

Find another input in the table that has more than one output.

Is this relation a function?
$x$ $y$
-43
-32
-21
-2-1
-10
0-1
1-2
13
2-3

Question 6. Click on .

Are there any inputs in the table that have two outputs (or are there two points with the same $x$-coordinate)?

Is this relation a function?
$x$ $y$
-54
-43
-32
-21
-10
0-1
10
21
32
43
54

Click on and use the table or the graph to decide if this relation is a function. Is it a function?
$x$ $y$
-51
-4-1
-31
-2-1
-11
0-1
11
2-1
31
4-1
51

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