Functions
5.1
Relations and Functions
In this section, you will learn about the relationship between relations and functions. Upon completion you will be able to:
• Translate between a set of real numbers and interval notation. • State whether or not a relation is a function.
• State the domain and range of a given graphical representation of a function, using interval notation. • Use and apply function notation to given scenarios.
Writing Interval Notation
We will focus on the use of the real numbers, R. Recall that we may visualize R as a line. Segments of this line are called intervals of numbers, which can be discussed symbolically or graphically.
Definition
• Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form {x | statement about x} which is read as, “ the set of all x such that the statement about x is true.” For example,
{x | 4 < x ≤ 12}
would be read as “the set of all x such that four is less than x is less than or equal to 12.”
• Interval notation is a way of describing sets that include all real numbers between a lower limit, that may or may not be included, and an upper limit, that may or may not be included. The endpoint values are listed, separated by a comma, between brackets or parentheses. A square bracket, “[” or “]”, indicates inclusion in the set, and a parenthesis, “(” or “)”, indicates exclusion from the set. The example given in set-builder notation above would be written, using interval notation, as
(4, 12].
N If the interval does not have finite endpoints, we use the symbol −∞ to indicate that the interval
extends indefinitely to the left and ∞ to indicate that the interval extends indefinitely to the right. Since infinity is a concept, and not a number, we always use parentheses when using these symbols in interval notation.
Below is a summary of the different ways to express given sets of numbers.
Set-Builder Segment of the Interval Verbal
Notation Real Number Line Notation Description
{x | a< x < b} a b
◦ ◦
(a, b) xis strictly between a and b
{x | a ≤ x< b} a b
• ◦
[a, b) xis between a and b and includes a
{x | a< x ≤ b} a b
◦ •
(a, b] xis between a and b and includes b
{x | a ≤ x ≤ b} a b
• •
[a, b] xis between a and b and includes a and b
{x | x< b} b
◦
(−∞, b) xis less than b
{x | x ≤ b} b
•
(−∞, b] xis less than or equal to b
{x | x> a} a
◦
(a, ∞) xis greater than a
{x | x ≥ a} a
•
[a, ∞) xis greater than or equal to a
{x | − ∞< x < ∞} (−∞, ∞) xis any real number
Example 1 Consider the sets of real numbers described below. Fill in the missing cells of the table.
Set Builder Notation Segment of Real Line Interval Notation Verbal Description
a. −4 6
◦ •
b. xis at most 5
c. {x | − 2 ≤ x ≤ 10}
We will often have occasion to combine intervals. There are two basic ways to combine intervals: intersection and union.
Definition
Suppose A and B are two intervals,
• The intersection of A and B, A ∩ B, is the portion of the real number line that the two intervals have in common.
• The union of A and B, A ∪ B, is the portion of the real number line which includes all points of both intervals.
Example 2 Express the following sets of numbers using interval notation. Draw number lines to help
demonstrate your answer. a. {x | x ≤ −4 or x ≥ 8} b. {x | x , 1} c. {x | x , ±2} d. {x | x ≤ 7 and x ≥ 0} e. {x | x ≥ −5 and x , 3}
Differentiating Between a Relation and a Function
DefinitionA relation is a set of ordered pairs in the coordinate plane.
The set consisting of the first components of each ordered pair is called the inputs and the set consisting
of the second components of each ordered pair is called the outputs.
As an example, consider the relation R= {(−7,4), (0,−6), (2,0)}. As written, R is described as a list. Since R consists of points in the plane, we can follow our instinct and plot the points.
x y −10 −5 5 10 −10 −5 5 10 For R we have Inputs: Outputs:
N For sets with a finite number of elements like these, the elements do not have to be listed in particular
One of the core concepts in this chapter and in calculus is the function. There are many ways to describe a function and we begin by defining a function as a special kind of relation.
Definition
A relation in which each x-coordinate is matched with only one y-coordinate is said to describe y as a
function of x.
In other words a function f is a relation that assigns a single element in the set of outputs to each element in the set of inputs, meaning no x-values are repeated.
p q r j m k Inputs Outputs Relation is a Function r q p j k Inputs Outputs Relation is a Function j m k p q Inputs Outputs Relation is NOT a Function
Two different inputs can be associated with one output, but one input cannot be associated with two different outputs.
All functions are relations, but not all relations are functions.
Definition
For any function,
• Each value in the domain of the function is also known as an input value, or independent variable, and is often labeled with the lowercase letter x.
• Each value in the range of the function is also known as an output value, or dependent variable, and is often labeled with the lowercase letter y.
To see what the function concept means geometrically, we graph two relations, R1and R2, in the
coordinate plane in the figures below.
R1= {(−2,1), (1,3), (1,4), (3,−1)} -5 -5 -4-4 -3-3 -2-2 -1-1 11 22 33 44 55 -2 -2 -1 -1 1 1 2 2 3 3 4 4 5 5 0 0 R2= {(−2,1), (1,4), (2,4), (3,−1)} -5 -5 -4-4 -3-3 -2-2 -1-1 11 22 33 44 55 -2 -2 -1 -1 1 1 2 2 3 3 4 4 5 5 0 0
Theorem 5.1 The Vertical Line Test
A set of points in the plane represents y as a function of x if and only if no two points lie on the same vertical line.
Practically speaking, the Vertical Line Test says that if we can drawn any vertical line that intersects a graph more than once, then the graph does not define a function and it fails the Vertical Line Test.
Example 3 Use the Vertical Line Test to determine which of the following relations describes y as a function of x. a. y= 3x x y −10 −5 5 10 −10 −5 5 10 b. x is always 4 x y −10 −5 5 10 −10 −5 5 10
To determine the domain and range of a function using its corresponding graph, we need to determine which x and y values occur as coordinates of points on the given graph.
Example 4 State the domain and range of the function given below, using interval notation.
x y • −10 −5 5 10 −10 −5 5 10
Example 5 State the domain and range of the function given below, using interval notation.
x y ◦ • ◦ −10 −5 5 10 −10 −5 5 10
Using Function Notation
There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.
Recall from the chapter on Linear Functions, we used standard function notation when representing the linear business models.
For example,
C(x)= mx + F
and
R(x)= px
Definition
The notation y= f (x) defines a function named f . This is read as “y is a function of x.”
The letter x represents the input value, or independent variable. The set of all input values is the domain of f . The letter y, or f (x), represents the output value, or dependent variable. The set of all output
values is the range of f .
Example 6 Use function notation to represent a function whose input is the date and the output is the
corresponding day of the week.
Example 7 A function N= f (b) gives the number of classrooms, N, in a specific building on campus,
b. What does f (Blocker)= 28 represent?
N The inputs to a function do not have to be numbers; function inputs can be names of people, labels of
geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with will have numbers as inputs and outputs.
When a table represents a function, corresponding domain and range values can also be specified using function notation.
Example 8 Use function notation, g(x)= y, to represent the function information given in each row of
the table below.
Domain Range
−2 3
0 1
7 15
When we know an input value and want to determine the corresponding output value for a function, defined algebraically, we evaluate the function using the function rule on the input value. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.
Example 9 Let f (x)= −x2+ x − 5.
Find and simplify the following: a. f (3)
c. f (x+ 3)
d. f (x)+ 3
! When simplifying −(2x)2, 2x is squared and then multiplied by −1, so that −(2x)2= −4x2. Also,
(a+ b)2= (a + b)(a + b) = a2+ 2ab + b2.
Be careful and follow the order of operations when evaluating functions: −(2x)2, (−2x)2
(a+ b)2, a2+ b2
Reflection:
• Can you represent a portion of the real number line using interval notation? • Can you explain what properties make a relation a function?
• Can you list the domain and range of a function from a graph? • What does f (∗) represent?
