2.1 Functions & Relations

Functions & Relations

A relation is a way of pairing quantities or objects; each pair consists of an input and an output. For example, the set {(1, 2), (1, 3), (2, 4), (5, 6)} represents a relation because each ordered pair has an x-value paired with a y-value. When you enter an input into a relation, it produces a certain output.

Think of a relation as a machine that pairs each input value with one or more output values.

​

A Function is a rule that assigns a unique output value to any input value

Multiple Choice

Identify the parts of this relation:

The input value is

• 5

• 20

• 100

This input-output machine
multiplies a number by 3.

This relation is shown in the table.

Every input value is paired with one particular output value. If you put "3" into the machine again and again, you will always get "9." That consistency makes this relation a function.

​​Some Relations are Functions

A function is a relation that pairs each input value with just one output value.

There are many real-world functions. One example is the formula that converts a temperature from degrees Fahrenheit to degrees Celsius. An input of 86°F always produces an output of 30°C. A web address (input) always takes you to the same web page (output). These relationships are both functions.

Now let's look at the relation in this table.

​A function pairs each input value with exactly one output value. In this table, the input value 8 has two different output values: $44 and $46. This relation is not a function.

A function cannot have more than one output for any input, but it can have the same output for several different inputs. An example is the absolute value function. Two different inputs, such as 3 and –3, can have the same output of 3, but each input has only one possible output.

​Identifying Functions

Multiple Choice

Which statement tells you that a relation is also a function?

• There are no duplicate input values.

• Every output value corresponds with only one input value.

• There is only one output value for every input value.

• The number of input values is the same as the number of output values.

Multiple Choice

Decide whether each relation is a function.

Input: Height of an object in feet
Output: Height of the same object in inches

• 
  Function

• Not
  Function

Multiple Choice

Decide whether each relation is a function.

Input: Hair Color
Output: Height

• Functions

• Not Functions

Multiple Choice

Decide whether each relation is a function.

Input: Number of cars in parking lot
Output: Number of blue cars in parking lot

• Function

• Not Function

Multiple Choice

Decide whether each relation is a function.

Input: Number of granola bars that cost $1.50 each
Output: Total cost

• Function

• Not Function

Representing Relations

Identify functions from tables, sets of ordered pairs, and mapping diagrams.

You can represent a relation as a table, a set of ordered pairs, or a mapping diagram.

In a relation, the input is often labeled x and the output is often labeled y.

One way to show these pairs is in a vertical table, with the x-values (inputs) on the left and the y-values (outputs) on the right. Horizontal tables may also be used, with the x-values on the top and the y-values below.

Another way to show x- and y-pairs of a relation is as a set of ordered pairs. Each row of a vertical table or column of a horizontal table represents an ordered pair, with the input (x) first, followed by the output (y).

• The vertical table represents the set of ordered pairs {(1, 3), (2, 6), (3, 9), (4, 12)}.

• The horizontal table represents the set of ordered pairs {(10, 30), (20, 60),
  (30, 90), (40, 120)}.

A third way to show a relation is with a mapping diagram, which shows how the inputs are paired with the outputs. Click the input/output values to see how the relation works.

Relations in Different Forms

You can represent a relation as a table, a mapping diagram, or a set of ordered pairs.

You can convert a table of values into a set of ordered pairs or a mapping diagram.

A relation pairs the following inputs and outputs: 2 with 0, 3 with 0, 5 with 4, and 7 with 8

The table shows the inputs represented as X-values and the outputs represented as y-values.

Table

You can use the information in a mapping diagram to create a set of ordered pairs or a table. To create a mapping diagram, place the x-values inside an oval on the left and the y-values in an oval on the right. Draw an arrow from each x-value to the corresponding y-value.

Mapping Diagram

​Order Pair are set of number in parenthesis. The 1st number ins you x values and the 2nd is you y values. (x,y)

​​Ordered Pairs

Multiple Select

The relation is Function or Not Function because

You need to have 2 boxes checked

• Function

• Not Function

• 
  there are fewer y-values than x-values

• 
  there is a single y-value for every x-value listed

• there is more than one x-value that results in a certain y-value

A set of ordered pairs represents a function when each x-value is paired with exactly one y-value. An x-value can be listed more than once, but it must always be paired with the same y-value. If two ordered pairs in a relation have the same x-values but different y-values, then the relation is not a function.

Identifying Functions

Multiple Choice

Which table of values represents a relation that is not a function?

The ordered pairs of points on the graph of a function represent the input/output pairs for that function.

Another way to see how x- and y-values in a relation are related is by graphing. Plotting the ordered pairs on a graph helps you determine whether the relation is also a function.

Each x-value is paired with exactly one y-value. That means this relation is a function.

Graphing Points

If an x-value in a relation is paired with more than one y-value, then the relation is not a function. In a graph, one of those points appears directly above the other, so you can draw a vertical line through the two points.

The vertical line test is a way of determining whether a relation is a function. If a vertical line can be drawn that passes through two or more points on the graph, then the relation is not a function.

The Vertical Line Test

Multiple Select

Use the vertical line test to select all graphs that represent functions.

A graph is a picture of a set of ordered pairs that can be plotted as points. Some graphs are shown as plotted points. Others are lines or curves.

​Look at this graph. It passes through some points whose x- and y-values are easy to identify. It also passes through other points whose x- and y-values are not easy to accurately identify.


All the plotted points are part of the relation. But between the plotted points, there are many more points on the graph. It is impossible to find the x- and y-values of every point on this graph in order to determine whether it represents a function. Fortunately, you can use the vertical line test to answer that question.

This graph passes the vertical line test. Any vertical line drawn on the graph passes through only one point. This graph represents a function.

Continuous Graphs

Multiple Select

Is this a graph of a function? Why?

You need to have 2 boxes clicked

• Yes

• No

• it fails the vertical line test

• it passes the vertical line test

Functions in Context

​The graph shows that this is a function. It passes the vertical line test. For each x-value there is only one y-value. Imagine a different scenario: What if there were two output values for one input? In that case, Bobby would haul passengers a certain number of miles (x) but wouldn't know which price to charge!

Find points on the graph. For each point, find the x-value along the bottom and the y-value along the left side. The first point has an x-value of 1 and a y-value of 6, so the ordered pair is (1, 6). The second point has an x-value of 3 and a y-value of 8, so the ordered pair is (3, 8).

Multiple Select

Which other ordered pairs are shown on the graph? Select all that apply.

• (4, 7)

• (7, 12)

• (5, 10)

• 
  (12, 7)

• (9, 4)

Multiple Select

Choose the two points that are on this graph.

• (11, 4)

• (4, 3)

• (2, 8)

• (7, 1)

• (1, 2)

Multiple Select

Is Sammy's graph a function? Why?

Make sure you have 2 answer click.

• Yes

• No

• it fails the vertical line test

• it passes the vertical line test