Download COL_4.1

HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Hawkes Learning Systems:
College Algebra
Section 4.1: Relations and Functions
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Objectives
o
o
o
o
Relations, domains and ranges.
Functions and the vertical line test.
Functional notation and function evaluation.
Implied domain of a function.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Relations, Domains and Ranges
Relations, Domains and Ranges
o A relation is a set of ordered pairs. Any set of
ordered pairs automatically relates the set of first
coordinates to the set of second coordinates, and
these sets have special names.
o The domain of a relation is the set of all the first
coordinates.
o The range of a relation is the set of all second
coordinates.
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Example 1: Relations, Domains and Ranges



The set R   4,2  ,  6, 1 ,  0,0  ,  4,0  ,  ,  2 is a
relation consisting of five ordered pairs.
What is the domain? R   4,2  ,  6, 1 ,  0,0  ,  4,0  ,  ,  2
4,6,0,  Note: it is not necessary to list – 4 twice in the domain.



HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Example 1: Relations, Domains and Ranges



The set R   4,2  ,  6, 1 ,  0,0  ,  4,0  ,  ,  2 is a
relation consisting of five ordered pairs.
2
What is the range? R   4, 2  ,  6, 1 ,  0, 0  ,  4, 0  ,  ,
2
2,

1,0,


 Again, you do not have to write 0 twice in the range.



HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example 2: Relations, Domains and Ranges
The equation 3x  7 y  13 describes a relation. This
relation consists of an infinite number of ordered pairs,
so it is not possible to list them all as a set.
Ex: One of the ordered pairs
of this relation is  2,1 ,
since  3 2   7 1  13. Both
the domain and range of this
relation are the set of real
numbers.
The graph of this relation is
shown to the right.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example 3: Relations, Domains and Ranges
The picture below describes a relation with infinite
elements.
What is the domain?  1,1
What is the range?  2,2
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example 4: Relations, Domains and Ranges
The picture below describes a similar relation to the
one in example 3. The shading indicates that all
ordered pairs inside the rectangle, as well as the
ordered pairs laying on the rectangle, are elements of
the relation.
What is the domain?  1,1
What is the range?  2,2
Note: The range and domain are the
same as in example 3. Does this mean
the relations are the same?
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example 5: Relations, Domains and Ranges
The set S   x, y  | x is the owner of y is a relation
among domestic dogs.
What is the domain? The set of all owners.
What is the range? The set of all dogs that are pets.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Functions and the Vertical Line Test
Functions
A function is a relation in which every element of the
domain is paired with exactly one element of the range.
Equivalently, a function is a relation in which no two
distinct ordered pairs have the same first coordinate.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Functions and the Vertical Line Test
Ex: Which of the following relations is a function?
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Functions and the Vertical Line Test
The Vertical Line Test
If a relation can be graphed in the Cartesian plane, the
relation is a function if and only if no vertical line
passes through the graph more than once. If even one
vertical line intersects the graph of the relation two or
more times, the relation fails to be a function.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Functions and the Vertical Line Test
Ex: Which coordinates of the relation
R   7,1 ,  3, 1 ,  0,0  ,  3,0  ,  5,2  fail the vertical
line test and, therefore, do NOT define the relation as
a function?  3, 1 (3,0)
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example 6: Functions and the Vertical Line Test
Consider the relation:
R   3,2  ,  1,0  ,  0,2  ,  2, 4  ,  4,0 
Do any vertical lines intersect the graph twice or more at any
one point?
No
Is the relation a function?
Yes
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example 7: Functions and the Vertical Line Test
Do any vertical lines intersect the graph twice or more
at any one point?
Yes
Is the relation below a function?
No
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example 8: Functions and the Vertical Line Test
Do any vertical lines intersect the graph twice or more
times at any one point?
No
Is the relation below a function?
Yes
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Functional Notation and Function Evaluation
Functional Notation
Functional notation is a means of writing a function in terms
of x.
Ex. Given the function 2 x  y  1, solve for y: y  2 x  1.
In this context, x is called the independent variable and y is
the dependent variable.
In functional notation, we would write the function above as
f  x   2 x  1 , read “f of x equals negative two x plus one.”
f  x   2 x  1 indicates that when given a specific value for x,
the function f returns negative two times that value, plus 1.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Functional Notation and Function Evaluation
Caution!
A common error is to think that f(x) stands for the
product of f and x. This is wrong! While it is true that
parentheses often indicate multiplication, they are also
used in defining functions.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Functional Notation and Function Evaluation
In defining the function f as f ( x)  2 x  1, the critical idea
being conveyed is the formula. We can use absolutely any
symbol as a placeholder in defining f.
For instance, f ( x)  2 x  1, f ( p)  2 p  1, f ( )  2 1
all define the same function.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Functional Notation and Function Evaluation
The variable, or symbol, that is used in defining a given
function is called its argument and serves as nothing more
than a placeholder.
Ex: What is the argument for each function?
f ( x)  2 x  1 The argument is x.
f ( p)  2 p  1 The argument is p.
f ( )  2 1 The argument is .
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example 9: Functional Notation and Function
Evaluation
The following equation in x and y represents a function.
Rewrite the equation in functional notation, then evaluate
for x = 5.
First, solve the equation for y.
12 x  4  3 y  2
12 x   3 y   6
3 y  12 x  6
y  4x  2
We can name the function any number of names.
f  x   4x  2
Here, we choose f, the same name in the previous
f  5  4  5  2
f (5)  22
examples.
We now evaluate f at 5 for the desired answer.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example 10: Functional Notation and
Function Evaluation
The following equations in x and y represent a function.
Rewrite the equations in functional notation, then evaluate
for x = – 2.
First, solve the
a. y  7  x 2
equation for y.
y  x2  7
2
g  x  x  7
2
g  2    2   7
g  2   4  7
g  2   11
Here we named
the function g.
We now evaluate
g at – 2 for the
desired answer.
Continued on the next slide…
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example 10: Functional Notation and
Function Evaluation (cont.)
The following equations in x and y represent a function.
Rewrite the equations in functional notation, then evaluate
for x = – 2.
Solve as usual.
b. 79  x  y  9
 y  9  79  x
y  9  79  x
h  x   9  79  x
h  2   9  79   2 
h  2   9  9
h  2   0
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Example 11: Functional Notation and
Function Evaluation
Given the function f  x   2 x 2  3, evaluate:
2
f
b

2
b
3
a.  
Simply replace x with b.
b. f  x  g   2  x  g   3
2
 2  x 2  2 xg  g 2   3
Here, we replace x with (x+g) and
simplify.
 2 x 2  4 xg  2 g 2  3
Continued on the next slide…
HAWKES LEARNING SYSTEMS
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Example 11: Functional Notation and
Function Evaluation (cont.)
Given the function f  x   2 x 2  3, evaluate:
Notice that the first term of the
f  x  g   f  x
numerator is the function we just
c.
simplified in b.
g
2 x 2  4 xg  2 g 2  3  2 x 2  3 4 xg  2 g 2


g
g
 4x  2g


HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Implied Domain of a Function
The domain of the function is implied by the formula
used in defining the function. It is assumed that the
domain of the function consists of all real numbers at
which the function can be evaluated to obtain a real
number: any values for the argument that result in
division by zero or an even root of a negative number
must be excluded from the domain.
HAWKES LEARNING SYSTEMS
math courseware specialists
Copyright © 2011 Hawkes Learning Systems.
All rights reserved.
Example 12: Implied Domain of a Function
Determine the domain of the following functions.
a. f  x   3x  7  x
Domain: x  7 or  ,7
x5
b. g  x   2
x 4
To avoid an even root of a negative number
we need to find x such that 7 – x ≥ 0. Solving
this inequality gives us the domain.
We want to avoid any x such that x 2  4  0,
as this would give us a denominator of zero.
After solving x 2  4  0 we get
 x  2  x  2   0 , so we define the domain
as excluding 2 and – 2.
Domain: x  2,2 or  , 2    2,2    2,  