Math worksheet
Functions
Relations
Definition of Functions
Function Notation
The Graph of a Function
Relations
A Relation is a set of ordered pairs
It may be specified in 4 ways
By a Graph
By mapping
By displaying the pairs
By an equation
Domain
Range
1
2
3
a
b
c
Domain
Range
Domain
Range
Input
Output
Domain
Range
Range
Domain
.
You may also use tables
| 1 | b |
| 2 | c |
| 3 | a |
A
B
Definition of Functions
A function is a relation in which each possible input value leads to exactly one output value.
It may be specified in 4 ways
By a Graph
By mapping
By displaying the pairs
By an equation
Input
Output
1
2
3
a
b
c
Input
Output
Input
Output
Output
Input
.
Ex. Determining If Menu Price Lists Are Functions
Item Price
Plain Donut ................................................ 1.49
Jelly Donut ................................................. 1.99
Chocolate Donut ........................................ 1.99
Is price a function of the item?
Is the item a function of the price?
Yes
No
Definition Functions
Ordered Pairs
Yes
Domain
Range
Yes
Domain
Range
No
Do the following relations define a function?
Find domain and range in the case the relation is a function.
No
Examples
Definition of Functions
Graph
A function is a relation in which each possible input value leads to exactly one output value.
Output
Input
Vertical Line Test
A graph represents a function if any vertical line drawn intersects the curve only once.
.
.
.
Domain
Range
The Domain is the input on the -axis
The Range is the output on the -axis
Letters used for functions include
Definition Functions
Graph
Domain
Range
Domain
Range
>
>
All real
numbers
All real
numbers
Not a Function
.
.
>
>
Is a Function
Is a Function
Examples
Not a Function
Definition of Functions
Equations
A function is a relation in which each possible input value leads to exactly one output value.
When you input one value of there is only one value for y
No
Yes
One input of yields
2 values of
Function Notation
The notation
defines a function named f.
is a function of
Input
Output
Instead of writing
we write
So is the name for the rule that defines
the output
Independent Variable
Dependent Variable
Independent Variable
Dependent Variable
.
Evaluating and Solving
Functions
Ordered pairs
Ex. If
Find: a.
b. Evaluate at 1.
Ex. Solve
For which is
From a table
Ex. Evaluate:
| 5 | 10 | 15 | 20 | 25 | 30 | |
| 10 | 9 | 8 | 7 | 6 | 5 |
Ex. Solve:
Evaluating and Solving
Functions
Evaluating and Solving
Functions
From a Graph
Independent
Variable
Dependent
Variable
.
Ex.
Evaluate: a.
b.
Evaluate: a.
Evaluating and Solving
Functions
From a Graph
>
>
Evaluate:
a.
b.
Solve:
a.
b.
Example
Evaluating and Solving
Functions
From an Equation
Find the value f(−2), where
Given the function ,
solve
+2
+2
A quadratic Equation
or
The value is -17
The Graph of a Function
is on the graph of
so is on the graph of
is always positive or 0
is all real numbers
is all positive numbers and 0
-intercept
-intercept
Understanding Graphs
>
>
For which values of is the function positive
Above the -axis
On
Example
For which values of is the function negative
Below the -axis
On
-intercept
-intercept
For which values of is the function above the line
On
Range
Domain
1) Find
2) Is positive or negative?
3) Find the -intercepts
4) Find the -intercept
5) Find where
6) Find where
7) How often does the line
intersect the graph of
8) For which values of is
Positive
3 times
and
Range
Domain
[-3.8, -3)