Download Define Function, Domain, and Range Identify Functions

Math 35
2.5 "An Introduction to Functions"
Objectives:
* De…ne function, domain, and range.
* Identify functions.
* Use the vertical line test.
* Use function notation.
* Find the domain of a function.
* Graph linear functions.
De…ne Function, Domain, and Range
De…nition:
Function
A function is a set of ordered pairs in which no two ordered pairs have
the same …rst coordinates and di¤erent second coordinates.
> The domain of a relation is the set of all …rst coordinates of the ordered pairs.
> The range of a relation is the set of all second coordinates of the ordered pairs.
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:
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:
Identify Functions
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
exactly one value of y in the range, then y is said to be a function of x.
In the previous de…nition, since y depends on x, we call x the independent variable and the y the dependent variable.
Example 2: (Determining if y is a function of x)
In each case, …nd the domain and range and determine whether the relation de…nes y to be a function of x.
a)
b)
Page: 1
x
y
3
5
0
2
3
1
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
2.5
Example 3: (Determining if y is a function of x)
Determine whether each equation de…nes y to be a function of x:
b) x2 + y 2 = 36
a) x + y = 5
Use the Vertical Line Test
The Vertical Line Test
kIf a vertical line intersects a graph in more than one point, the graph is not the graph of a function.k
Example 4: (Using the vertical line test )
Determine whether each of the following is the graph of a function.
y
2
a)
y
b)
-2
2
-2
1
-4 -2
x
2 4
x
-1
Use Function Notation
Function Notation:
The notation
For example:
WARNING!
indicates that y is a function of x:
f (x) = 2x
5
The symbol f (x) denotes a function.
Example 5: (Finding the output of a function)
x2 + 2x
Let f (x) =
: Find:
2
a) f (2)
It does not mean f x
(f times x)
b) f (w + 2)
Page: 2
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
2.5
Example 6: (Finding the output of a function)
Let g (x) = 2x2
5x
4: Find:
b) g t2
a) g ( 4)
1
Example 7: (Finding the input when the output is giving)
x
Let f (x) = + 4: For what value of x is f (x) = 5?
3
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 8:
Find the domain of each function.
a) f (x) = 2x
6
b) g (t) =
2
t2 + 3t + 2
Page: 3
c) h (s) = js
3j
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
2.5
Graph Linear Functions
A linear function is a function that can be written in the form
f (x) = mx + b
:
Its graph is a straight line with
slope m and y intercept (0; b) :
De…nition:
"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).
Example 9: (Graphing a line)
Graph:
f (x) =
3x
2
y
4
x
f (x) =
3x
2
2
-4
-2
2
-2
4
x
-4
Page: 4
Notes by Bibiana Lopez