#### Math - 2018-19

# 8.15 - Domain and Range

8.15The student willa)

determinewhether a given relation is a function;b)

determinethe domain and range of a function.

*Adopted: 2016*

### BIG IDEAS

- I can compare the cost of pizza with number of toppings, find
the correlation between number of hours spent in the sun and development of
skin cancer, match the amount of money put into a vending machine with the kind
of selection that comes out, and equate the growth of plants with amount of
food and water they receive.

- I will recognize that as the
value of the dependent variable changes the independent variable also changes.

### UNDERSTANDING THE STANDARD

· A relation is any set of ordered pairs. For each first member, there may be many second members.

· A function is a relation between a set of inputs, called the domain, and a set of outputs, called the range, with the property that each input is related to exactly one output.

· As a table of values, a function has a unique value assigned to the second variable for each value of the first variable. In the “not a function” example, the input value “1” has two different output values, 5 and -3, assigned to it, so the example is not a function.

·

As a set of ordered pairs, a function has a unique or different

*y*-value assigned to each

*x*-value. For example, the set of ordered pairs, {(1, 2), (2, 4), (3, 2), (4, 8)} is a function. This set of ordered pairs, {(1, 2), (2, 4), (3, 2), (2, 3)}, is not a function because the

*x*-value of “2” has two different

*y*-values.

·
As a graph
of discrete points, a relation is a function when, for any value of *x*, a vertical line passes through no
more than one point on the graph.

· Some relations are functions; all functions are relations.

· Graphs of functions can be discrete or continuous.

· In a discrete function graph there are separate, distinct points. You would not use a line to connect these points on a graph. The points between the plotted points have no meaning and cannot be interpreted. For example, the number of pets per household represents a discrete function because you cannot have a fraction of a pet.

· Functions may be represented as ordered pairs, tables, graphs, equations, physical models, or in words. Any given relationship can be represented using multiple representations.

· A discussion about determining whether a continuous graph of a relation is a function using the vertical line test may occur in grade eight, but will be explored further in Algebra I.

·
The domain
is the set of all the input values for the independent variable or *x*-values (first number in an ordered
pair).

·
The range is
the set of all the output values for the dependent variable or *y*-values (second number in an ordered
pair

·
If a
function is comprised of a discrete set of ordered pairs, then the domain is
the set of all the

*x*-coordinates, and the range is the
set of all the *y*-coordinates. These sets of values can be determined given
different representations of the function.

- Example: The domain of a function is {−1, 1, 2, 3} and the range is {−3, 3, 5}. The following are representations of this function:

o

The function represented as a table of values:

o The function represented as a set of ordered pairs: {(-1, 5), (1, -3), (2, 3), (3, 5)}

o The function represented as a graph on a coordinate plane:

### ESSENTIALS

- What are the similarities and differences among the terms domain, range, independent variable and dependent variable? The value of the dependent variable changes as the independent variable changes. The domain is the set of all input values for the independent variable. The range is the set of all possible values for the dependent variable.

**The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to**

·
8.15a_{1}** Determine**
whether a relation, represented by a set of ordered pairs, a table, or a graph
of discrete points is a function. Sets are limited to no more than 10 ordered
pairs.

·
8.15b_{1}** Identify**
the domain and range of a function represented as a set of ordered pairs, a
table, or a graph of discrete points.

### KEY VOCABULARY

domain, range, independent variable, dependent variable, function

*Updated: Nov 20, 2018*