Basics of Functions &
Their Graphs
Section 1.2
Created by Laura Ralston
Relation
 Is
any set of ordered pairs that
represents some relationship
 Examples:
Naming children of parent x
 Birthdays
 Convert yards to feet
 Fuel efficiency and Time
 Sale price of an item discounted 25% with an
original price x
 Square of a number x
 Activity and Calories Burned per hour

Relation

Domain: set of all
first components (xvalues) or inputs of
the ordered pairs

Range: set of all
second components
(y-values) or
outputs of the
ordered pairs
What is a function?
 Functions
are a special relation that can
be used to model important phenomena
in our world
 An important concept in mathematics
 Each valid input always determines
exactly one output!!!
 Input
is typically “x” or the independent
variable
 Output is typically “y” or dependent
variable
How do I know if it is a
function?
A
function is a relation in which no two
ordered pairs have the same first
component and different second
components
 In short, no repeat of x-values
Function Notation
 Emphasizes
y
that y is a function of x
= f(x)
 read
as “y equals f of x”
 does NOT mean to multiply f and x
 denotes function f with input x produces
output y
 f(input)
= output
Function Representations
 VERBAL
 NUMERICAL
(Table of Values)
 DIAGRAMMATIC
 GRAPHICAL
 SYMBOLIC
VERBAL
Words describe
precisely what is
computed
 May be oral or
written
 May be in
algorithmic form
where each step is
precisely stated


Examples

To calculate the
square of a number
x, multiply x by x to
obtain y.
1. Input a number x
 2. Multiply x by x;
Call this result y.
 3. Output y, the
square of a number

NUMERICAL (Table of Values)
Lists of input-output
pairs
 May be in the form
of a table or an
explicit set of
ordered pairs
 Difficult to
impossible to list ALL
possible inputs


partial numerical
representation

Provides Limited
Explicit Information

(1,1) (2,4) (3,9)
(4,16) (5,25)

Table
DIAGRAMMATIC
 Shows
inputs and outputs visually
 Uses no words, formulas, or tables
 Difficult to impossible to show every
input
 Provides Limited Explicit Information
GRAPHICAL
 First
used by Leonard Euler, who also
invented function notation
 Visually pairs x-input with a y-output
 Plotted in the xy-plane
 Provides Complete Explicit Information
for a given interval
SYMBOLIC

Mathematical
Formula is an
efficient and concise
way of representing
a function

Provides Complete
Implicit Information
at any given point
Implied Domain is
the set of all valid
inputs that make
sense
 F(x) = x2
 g(x) = 3x
 h(x) = x - .25x =
.75x
 f(x) = x/5

Which representation is best ?
 Depends
on the questions that need to
be answered, the type of information to
be shared, and the audience
 Most newspapers, magazines, and TV
reports use tables and graphs so that
information can be absorbed easily
 Scientific journals use formulas since
their concern is precision

Every function is a
relation, but NOT
every relation is a
function

Two things to
determine:

Is the relation a
function?
Stronger statement
 Unique output for each
input


If so, what is the
domain and range?
 You
need to know the domain and
range because functions can fail to yield
reasonable results if
 input
is not physically reasonable
 output is not physically reasonable
 output is not mathematically reasonable
Helps to avoid “nonsense” answers
Is it a function? If so, domain
and range?
 Given
a numerical model or a diagram
ask yourself “is each x-value paired
with exactly one y-value?”
 Domain
is input- x - left column or top row,
left oval in mapping
 Range is output - y- right column or
bottom row, right oval in mapping
 EXAMPLES
 Given
a coordinate plane graph, use
vertical line test
 Visualize
vertical lines in the xy plane
 If each vertical line intersects the graph at
NO more than one point, then it is a
function.
 Domain - “walk along x-axis” - look up and
down - do you see the graph?
 Range - “walk along y-axis” -look left and
right
 Given
a symbolic representation, create
the graph and use the vertical line test.
 Determine DOMAIN only  Two
restrictions!!!
 Dividing
by Zero
 Taking an even indexed root of a negative
number
 Examples
Technology is usually more efficient and reliable
than paper and pencil