Download 2.5

Math 35 (Spring ’09)
2.5 "An Introduction to Functions"
—————————————————————————————————————————————————
Objectives:
*
Identify functions and determine domain and range
*
Use function notation
*
Find the domain of a function
*
Use the vertical line test
—————————————————————————————————————————————————
De…ne Relation, Domain, and Range & Identify Functions
The concept of a function is one of the most important ideas in all of mathematics.
De…nition:
"Function, Domain, and Range"
A function is a set of ordered pairs (a relation) in which to each
_____________________
there corresponds exactly one ______________________________.
The set of …rst components is called
___________________________.
The set of second components is called
__________________________.
Since we will often work with sets of ordered pairs of the form (x; y), it is helpful to de…ne a function
using the variables x and y:
Determining if " y is a Function of x
Given a relation in x and y, if to each value of x in the domain there corresponds
__________________ value of y in the range, then y is said to be a function of x.
Note:
In the previous de…nition, since y depends on x, we call:
- x ____________________________.
- y ____________________________.
Example 1: (Finding the domain and range)
Find the domain and range of the relation f(5; 6) ; ( 12; 4) ; (8; 6) ; ( 6; 6) ; (5; 4)g
Page: 1
Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
2.5
Example 2: (Determine if y is a function of x)
In each case, determine whether the relation de…nes y to be a function of x.
a)
b)
WARNING!
Question:
x
y
3
5
0
2
3
1
Not all equations de…ne functions.
What kind of linear equations give you functions?
Example 3: (Determine if y is a function of x)
Determine whether each equation de…nes y to be a function of x:
a)
y=
1
x2
b)
y
2
=x
Use Function Notation
Function Notation:
kThe notation __________________ indicates that y is a function of x:k
For example:
WARNING!
f (x) = 2x
5
The symbol f (x) denotes a function.
It does not mean f x
Page: 2
(f times x)
Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
2.5
Example 4: (Finding the output when the input is giving)
Let f (x) =
a)
x2 + 2x
:
2
Find:
f (2)
b)
f ( 1)
Example 5: (Finding the input when the output is giving)
Let f (x) =
x
3
+ 4: For what value of x is f (x) = 5
Find the Domain of a Function
We can think of a function as a machine that takes some input x and turns it into some output f (x).
The set of numbers that we put into the machine is the domain of the function,
and the set of numbers that comes out is the range.
Example 6:
Find the domain of each function:
a)
f (x) = 2x
6
b)
g (t) =
2
t+3
Page: 3
c)
h (s) = js + 3j
Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
De…nition:
2.5
"Identity Function"
The most basic linear function ________________ is called
the identity function because it assigns each real number to itself.
De…nition:
"Constant Function"
A linear function de…ned by _______________ is called a constant function,
because for any input x, the output is the constant b: (horizontal line).
Use the Vertical Line Test
Some graphs de…ne functions and some do not. To …nd if a graph is a function we will use the vertical line test.
The Vertical Line Test
kIf a vertical line intercepts a graph in more than one point, the graph is not the graph of a function.k
Example 7: (Using the vertical line test )
Determine whether each of the following is the graph of a function.
y
a)
1.0
y
0.5
2
-2
2
b)
x
-4
-2
-0.5
2
4
x
-2
-1.0
Page: 4
Bibiana Lopez