SECTION 1.1SECTION 1.1
FUNCTIONSFUNCTIONS
DEFINITION OF A RELATION
DEFINITION OF A RELATION
A relation is a correspondence A relation is a correspondence between two sets. If x and y between two sets. If x and y are two elements in these sets are two elements in these sets and if a relation exists between and if a relation exists between x and y, we say that x x and y, we say that x corresponds to y or that y corresponds to y or that y depends upon x. The depends upon x. The correspondence can be written correspondence can be written as an ordered pair (x,y).as an ordered pair (x,y).
DEFINITION OF A RELATION
DEFINITION OF A RELATION
Thus, a relation is simply a set Thus, a relation is simply a set of ordered pairs or a table of ordered pairs or a table which relates x and y values.which relates x and y values.
DEFINITION OF FUNCTION
DEFINITION OF FUNCTION
Let X and Y be two nonempty Let X and Y be two nonempty sets of real numbers. A sets of real numbers. A functionfunction from X into Y is a rule from X into Y is a rule or a correspondence that or a correspondence that associates with each element of associates with each element of X a X a uniqueunique element of Y. This is element of Y. This is a special type of relation.a special type of relation.For every x, there is only For every x, there is only
one y!one y!
DOMAIN AND RANGEDOMAIN AND RANGE
The set of all x The set of all x values.values.
DEFINITIODEFINITIONNOFOF
RANGERANGE
The set of all y The set of all y values.values.
Also called Also called “functional “functional values”.values”.
DEFINITIODEFINITIONNOFOF
DOMAINDOMAIN
THE FUNCTION AS A “MAPPING”
THE FUNCTION AS A “MAPPING”
11
44
77
--22
88
22
00--33
x-x-valuesvalues
y-y-valuesvalues
DOMAIDOMAINN
RANGERANGE
Ordered Ordered Pairs:Pairs:(1 , (1 ,
2)2)(4 , (4 , 8)8)(7, - (7, - 3)3)(- 2, (- 2, 0)0)
THE FUNCTION AS A “MAPPING”
THE FUNCTION AS A “MAPPING”
FrankFrank SueSue
AA
BB
CC
JillJill
Consider 3 students whose Consider 3 students whose names are mapped to their letter names are mapped to their letter grades on the last History exam:grades on the last History exam:
For each person For each person in the domain, in the domain, there can be there can be only one only one associated associated letter grade in letter grade in the range.the range.
THE SQUARING FUNCTION
THE SQUARING FUNCTION
--22--1100
11
00 11 44 99
22 33
Each Each element in element in the domain the domain maps to its maps to its square.square.
COUNTER-EXAMPLE:COUNTER-EXAMPLE:
55
11 22 33
44Ordered Pairs:Ordered Pairs:
(4, 1)(4, 1)
(4, 2)(4, 2)
(5, 3)(5, 3)
This is an example of a This is an example of a relation but not a function.relation but not a function.
THREE WAYS TO REPRESENT A FUNCTION
THREE WAYS TO REPRESENT A FUNCTION
NUMERICALLY - ordered NUMERICALLY - ordered pairspairs
SYMBOLICALLY - equationSYMBOLICALLY - equation
GRAPHICALLY - pictureGRAPHICALLY - picture
EXAMPLEEXAMPLE
Determine whether the Determine whether the relation represents a function:relation represents a function:
(a) {(1,4),(2,5),(3,6),(4,7)}(a) {(1,4),(2,5),(3,6),(4,7)}
EXAMPLEEXAMPLE
Determine whether the Determine whether the relation represents a function:relation represents a function:
(a) {(1,4),(2,4),(3,5),(6,10)}(a) {(1,4),(2,4),(3,5),(6,10)}
EXAMPLEEXAMPLE
Determine whether the Determine whether the relation represents a function:relation represents a function:
(a) {( - 3,9),(- 2,4),(0,0),(1,1),(a) {( - 3,9),(- 2,4),(0,0),(1,1),( - 3,8)}( - 3,8)}
EXAMPLEEXAMPLE
Determine whether the Determine whether the relation represents a function:relation represents a function:
(a) {((a) {( - 3- 3,9),(- 2,4),(0,0),(1,1),,9),(- 2,4),(0,0),(1,1),(( - 3- 3,8)},8)}
EVALUATING A FUNCTION AT A GIVEN X-VALUE
EVALUATING A FUNCTION AT A GIVEN X-VALUE
f(x) f(x) = =
f(0) f(0) = =
f(2) f(2) = =
f(-2) f(-2) ==
f(9) f(9) = =
x x 22
00
44
44
8181
0 00 0
2 42 4
- 2 4- 2 4
9 819 81
xx f(x) f(x)
Symbolically, the squaring Symbolically, the squaring function can be represented asfunction can be represented as
y = xy = x 2 2
““FUNCTIONAL NOTATION”FUNCTIONAL NOTATION”
f(x) = xf(x) = x 2 2
Read:Read: “f of x equals x “f of x equals x squared”squared”
EVALUATING A FUNCTION AT A GIVEN
X-VALUE
EVALUATING A FUNCTION AT A GIVEN
X-VALUEFor f(x) = 2x2 – 3x, find
the values of the following:
(a) f(3) (b) f(x) + f(3)(c) f(-x)
(d) - f(x) (e) f(x + 3)
(f)
hf(x)h)f(x
FINDING VALUES OF A FUNCTION ON A
CALCULATOR
FINDING VALUES OF A FUNCTION ON A
CALCULATOR
DO EXAMPLE 7
IMPLICIT FORM OF A FUNCTION
IMPLICIT FORM OF A FUNCTION
When a function is defined by an equation in x and y, we say that the function is given implicitly. If it is possible to solve the equation for y in terms of x, then we write y = f(x) and say that the function is given explicitly. See examples on Pg 127.
DETERMINING WHETHER AN EQUATION IS A
FUNCTION
DETERMINING WHETHER AN EQUATION IS A
FUNCTION
Determine if x2 + y2 = 1 is a function.
2x1y
This means that for certain values This means that for certain values of x, there are two possible of x, there are two possible outcomes for y.outcomes for y.Thus, this is not a function!Thus, this is not a function!
Important Facts About Functions:
Important Facts About Functions:
1.1. For each For each xx in the domain of a in the domain of a function function ff, there is one and only , there is one and only one image one image f(x)f(x) in the range. For in the range. For every every xx, there is only , there is only one one yy..
2.2. ff is the symbol we use to is the symbol we use to denote the denote the function. It is symbolic function. It is symbolic of the of the equation that we use to equation that we use to get from an get from an xx in the domain to in the domain to f(x)f(x) in the range. in the range. f(x)f(x) is another is another name for name for yy..
Important Facts About Functions:Important Facts About Functions:
3.3. If y = If y = f(x)f(x) , then , then xx is called the is called the independent variable or argument independent variable or argument of of ff, and , and yy is called the dependent is called the dependent
variable or the value of variable or the value of ff at at xx..
DOMAIN OF A FUNCTION
DOMAIN OF A FUNCTION
If a function is being If a function is being described symbolically and it described symbolically and it comes with a specific domain, comes with a specific domain, that domain should be that domain should be expressly given.expressly given.
Otherwise, the domain of the Otherwise, the domain of the function will be assumed to function will be assumed to be the “natural domain”.be the “natural domain”.
EXAMPLE: f(x) = x 2EXAMPLE: f(x) = x 2
Knowing the function of Knowing the function of squaring a number, we can squaring a number, we can determine that the natural determine that the natural domain is all real numbers domain is all real numbers because any real number can because any real number can be squared.be squared.
We can also look at a graphWe can also look at a graph..
EXAMPLE: f(x) = x 2 + 5x
EXAMPLE: f(x) = x 2 + 5x
This is simply a modification This is simply a modification of the squaring function. of the squaring function. Thus, we can determine that Thus, we can determine that the natural domain is all real the natural domain is all real numbers.numbers.
We can also look at a graphWe can also look at a graph..
EXAMPLEEXAMPLE
Find the Find the domain:domain: 42x
3x = f(x)
D: { xD: { x } }
2x
2x
EXAMPLEEXAMPLE
Find the Find the domain:domain:
3t-4 = f(t)
4 – 3t 0 4 – 3t 0
3t 43t 4
34
t
EXAMPLEEXAMPLE
Find the domain Find the domain ::
f(x) = xf(x) = x33 + x - + x - 11
All real numbersAll real numbers
OPERATIONS ON FUNCTIONS
OPERATIONS ON FUNCTIONS
Notation for four basic Notation for four basic operations on functions:operations on functions:
(f + g)(x) = f(x) + g(x)(f + g)(x) = f(x) + g(x) (f - g)(x) = f(x) - g(x)(f - g)(x) = f(x) - g(x) (f (f g)(x) = f(x) g)(x) = f(x) g(x)g(x) (f / g)(x) = f(x) / g(x)(f / g)(x) = f(x) / g(x)
OPERATIONS ON FUNCTIONS
OPERATIONS ON FUNCTIONS
Do Example 10Do Example 10
CONCLUSION OF SECTION 1.1CONCLUSION OF SECTION 1.1