1.3.2 Functions and Their Representations II
Representation 2: Defining a function using a
diagram
Function name:
Definition:
mother
Such a diagram is often referred to as a mapping
diagram.
Rule of association:
an input will be at the tail of the arrow
the associated output will be at the head of the arrow
Domain: {Sam, Maria, Jennifer}
Range: {Gertrude, Josephine}
Examples of usage:
we say:
we write:
input: Sam
"mother of Sam is Gertrude"
mother(Sam) = Gertrude
output: Gertrude
1.3.2-1
Representation 3: Defining a function as a set of
ordered pairs
Example:
name: mother
definition:
mother = {(Sam, Gertrude), (Maria, Gertrude), (Jennifer, Josephine)}
what is the rule of association here?
mother(Maria) = Gertrude
Domain: {Sam, Maria, Jennifer}
Range: ?
1.3.2-2
Representation 4:
Defining a function numerically (by table)
Function name: g
Definition:
x
1
3
4
0
g(x)
5
-2
5
0
Rule of association:
inputs are in top row of table
corresponding outputs are below in second row
Examples of usage:
g(1) =
g(0) =
g(2) =




a function defined symbolically
can also be represented numerically
this will be only a partial representation
since a function defined symbolically has an infinite
domain
 therefore, not every input-output pair can be tabulated
Example: f(x) = x2
Partial numeric representation:
x
-2
-1
0
f(x)
4
1
2
1.3.2-3
Representation 5: Defining a function using an
equation
This method is not specifically mentioned in the book, but
it makes some other things easier to explain.
Example:
Name:
p
Definition:
the function p defined by equation: y = x2
Rule of association:
when x and y are the variables, implicitly assume:
x stands for an input
y stands for an output
the equation is used to associate an input with an output
input variable x is called the independent variable
output variable y is called the dependent variable
(because its value depends on what value
was chosen for x)
Example of usage:
p(0) = ??
p( 3 ) = ??
Note: p is also defined symbolically by: p(x) = x2
Example: Celsius to Fahren+heit function:
by formula:
by equation:
F(C) = (9/5)C+32
the function F (with independent
variable C) defined by the equation
F = (9/5)C + 32
1.3.2-4
Some equations don’t define functions
equation:
y2 = 1 + x
for input x = 3 what is the output?
there would be two outputs: y = 2 or y = -2
This is a NO-NO for functions!
For a function, every input must have exactly one output
So the above equation does not define a function.
Function or not? and why?
You may be given a relation in any of the representations
1-5, and asked if it represents a function, and explain why
or why not. Here is how you are to respond:
If the relation is not a function, state an input that has two
outputs, and state what they are. Then state “has an input
with two outputs, therefore not a function”.
If the relation is a function, state “each input has exactly
one output” (don’t forget the exactly).
Example:
Question: Does y2 = x + 8 define a function?
Answer: if x = 8, y = 4 or -4. Has an input with two
outputs, therefore not a function.
Terminology:
 for y = 1 + x2 (which is an equation that defines a
function with independent variable x)
 we say that y is a function of x
 on the other hand, x is not a function of y
1.3.2-5