Characteristics & Properites of FUNctions
(Domain and Range)
What is a function?
A function is a relation in which there is a unique output for each input.
Each value of the independent variable (the domain) must correspond
to only one value of the dependent variable (the range).
INPUT
OUTPUT
DOMAIN
Ex 
f(x)
RANGE
Which of the following relations represent functions?
D = { Mississauga phone numbers }
R = { Mississauga residences/businesses }
INPUT:
(905)824–1025
Function?
OUTPUT:
D = { eye colours }
R = { names of people in class }
INPUT:
brown
Function?
OUTPUT:
Functions can be represented graphically, numerically, or algebraically.
How can you determine if a relation is a function in each form?
Examples
GRAPHICALLY
 scatter plot
 graph
Function? Use the vertical line
test (VLT).
NUMERICALLY
 set of ordered pairs
 table of values
 mapping diagram
Check that each value
of the domain
corresponds to only
one value of the
range.
ALGEBRAICALLY
 equation
Use reasoning
(consider restrictions
on the domain and/or
range).
Ex 
(Review graphical and numerical examples from the previous lesson.)
Ex 
Determine which of the following equations represent functions.
State the domain and range of each relation by using reasoning.
a)
y = –2(x – 1)2 + 3
b)
y=
c)
y=
d)
x2 + y2 = 9
e)
y = 2x
f)
y = sinx – 1
g)
y = 2cosx
1
x2
1
x2
Ex 
Compare y = x2 to x = y2. Which relation is a function?
Ex 
Barney rides a ferris wheel that has a diameter of 6 m. The axle of the
ferris wheel is 4 m above the ground. The ferris wheel takes 90 s to
make one complete revolution, and Barney rides for 10 rotations.
Determine the domain and range of the function that models Barney’s
ride on the ferris wheel.
D = { time on the ferris wheel in s }
R = { height above the ground in m }