Transcript Document

Definition of Function
or
How to Relate
Definition of a Relation
Relation
(A)
(B)
(C)
(1) 32 mpg
(2) 8 mpg
(3) 16 mpg
Domain and Range
• The values that make up the set of
independent values are the domain
• The values that make up the set of
dependent values are the range.
• State the domain and range from the 4
examples of relations given.
Domain
Correspondence
or
Relation
Range
Definition of a Relation
• A Relation maps a value from the
domain to the range. A Relation is a
set of ordered pairs.
• The most common types of relations in
algebra map subsets of real numbers to
other subsets of real numbers.
Example
Domain
Range
3
π
11
-2
1.618
2.718
Definition of a Function
• If a relation has the additional
characteristic that each element of the
domain is mapped to one and only one
element of the range then we call the
relation a Function.
Definition of a Function
• If we think of the domain as the set of
boys and the range the set of girls, then
a function is a monogamous
relationship from the domain to the
range. Each boy gets to go out with one
and only one girl.
• But… It does not say anything about the
girls. They get to live in Utah.
Decide if the Relation is a
Function.
• The relation is the year and the cost of a first
class stamp.
• The relation is the weight of an animal and
the beats per minute of it’s heart.
• The relation is the time of the day and the
intensity of the sun light.
• The relation is a number and it’s square.
• The relation is time since you left your house
for work and your distance from home.
Examples Please
• Give three examples from the real world
of relations. Be sure and state the
domain, the range, and the definition of
how the variables are related.
• Decide which if any of your examples
are functions.
NOT A FUNCTION
R
x
y1
y2
DOMAIN
RANGE
FUNCTION
f
x1
y
x2
DOMAIN
RANGE
Mathematical Examples
• Decide if the following relations are
functions.
X Y X Y
X Y
X Y
1
2
-5 7
1
1
1
2
-5 1
1
7
-1 2
-1 1
1
2
3
3
1
3
3
1
1
π
π
1 -1
5
Ways to Represent a Function
• Symbolic
x,y y  2x
or
y  2x
• Numeric X Y
1
2
5 10
-1 -2
3
• Graphical
6
• Verbal
The cost is twice
the original
amount.
Function Notation
The Symbolic Form
• A truly excellent notation. It is concise
and useful.
y  f x 
y  f x 
Name of the
function
• Output Value
• Member of the Range
• Dependent Variable
• Input Value
• Member of the Domain
• Independent Variable
These are all equivalent
names for the y.
These are all equivalent
names for the x.
Examples of Function
Notation
• The f notation
f x   3x  x
2
gx   x  4 x  1
2
• Find
f(2), g(-1), f(-0.983),
Your Turn!
x 1
Given: f x  
2x  3
Evaluate the following:
a  f 1
b f 2
c f 1.5
d  f a 
Graphical Representation
• Graphical representation of functions
have the advantage of conveying lots of
information in a compact form. There
are many types and styles of graphs but
in algebra we concentrate on graphs in
the rectangular (Cartesian) coordinate
system.
Graphs and Functions
Range
Domain
Determine the Domain and
Range for Each Function
From Their Graph
Vertical Line Test for
Functions
• If a vertical line intersects a graph once
and only once for each element of the
domain, then the graph is a function.
How to determine Domain and
Range of a function.
• Graph the following on your calculator.
Also give the algebra.
x
a f x 
x3
c 
1
f x  
x
2

b f x  x
3
 5x
Big Deal!
• A point is in the set of
ordered pairs that make up
the function if and only if
the point is on the graph of
the function.
Key Points
• Definition of a function
• Ways to represent a function
Symbolically
Graphically
Numerically
Verbally