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The Function Concept
• DEFINITION: A function consists of two
nonempty sets X and Y and a rule f that
associates each element x in X with one
and only one element y in Y.
Read “The function f from X into Y” and
symbolized by
f : X  Y.
The function f from X into Y
X
f
F “maps” X into Y
Y
Some examples:
•Supermarket item  price
•Student  chair
•College student  GPA
•Worker  SSN
•Car  license plate “number”
•Real number x  x2
More examples: Are these functions???
X

Y
• Dormitory rooms
Students
Rule: room  student(s) assigned
• Airplane luggage
Passengers
Rule: piece(s) of luggage  passenger
• Nine digit numbers
Workers
Rule: number  worker’s SSN
• Real numbers
Real numbers
Rule: x  the numbers y such that y2= x
Another defintion:
Let X and Y be sets. A function f from X
into Y is a set S of ordered pairs (x,y),
x X, y Y, with the property that (x1, y1) and
(x1, y2) are in S if and only if y1 = y2.
Examples
(1) S  {( 0,1), (1,2), (1,2), (2,5), (2,5), (3,10), (3,10)}
(2) S  {( 0,0), (1,1), (2,1), (1,2), (3,4), (3,6)}
Some Terminology & Notation
Let f : X  Y.
The set X (the “first” set) is called the domain
of the function.
The set of y’s in Y which correspond to an
element x in X is called the range of the
function. The range of f is, in general a
subset of Y.
Variables:
Let f : X  Y.
The symbols x and y are called variables.
In particular, a symbol such as x, representing
an arbitrary element in the domain is called an
independent variable.
A symbol such as y, representing an element in
the range corresponding to an element x in the
domain is called a dependent variable.
Function notation:
Let f : X  Y.
Pick an element x in X and apply the rule f.
This produces a unique element in Y. The
symbol f(x) is used to denote that element.
f(x) is read “f of x” or “the value of f at x”
or “the image of x under f .
Another picture
X
x
Y
f
 f(x)
More pictures
Y
X
f
f(X)
“Black box”
x
f
f(x)
One-to-one functions:
Let f : X  Y.
f is a one-to-one function if it takes distinct
elements in the domain to distinct elements in the
range. That is: f is one-to-one if
x1  x2
implies
Notation: f is 1 – 1.
f(x1)  f(x2).
Examples: Which of these function is 1 – 1?
• Supermarket item  price
• Student  GPA
• Car  license plate “number”
• f(x) = 2x + 3
Inverse functions
Suppose f : XY is 1 – 1. Then there is a function
g: f(X)X such that g(f(x)) = x for all x  X.
f
X
Y
f(X)
g
g is called the inverse of f and is denoted by f -1
Functions in Mathematics
•
From Geometry and Measurement:
1. Length function: x is a line segment,
l(x) = the length of x.
2. Area functions: x is a rectangle,
A(x) = the area of x.
3. Volume functions: x is a sphere,
V(x) = the volume of x.
•
From Probability & Statistics:
E is a subset (event) in a sample space S,
P(E) = the probability that E “occurs”.
Functions in “Algebra”
Let f : X  Y where X is a given set of real
numbers and Y is the set of all real numbers.
“f is a real-valued function of a real variable”
Note: The domain X may or may not be the set
of all real numbers.
Examples:
Graph of a function
Let f : X  Y. The graph of f is the set of
points (x, f(x)) plotted in the coordinate
plane:
Graph of f = {(x, f(x)) | x X }.
The graph of f is a “geometric” object – a
“picture” of the function.
Examples:
f ( x)  x 2  1
f ( n)  n 2  1
f ( x)  x 2  1
f ( x)  x 2  1
Functions defined on the positive integers:
Sequences
A function f whose domain is the set of
positive integers is called a sequence.
The values f (1), f (2), f (3), , f (n),
are called the terms of the sequence; f(1) is
the 1st term, f(2) is the 2nd term, and so on
Subscript notation
It is customary to use subscript notation
rather than functional notation:
f (1)  a1 , f (2)  a2 , f (3)  a3 ,
, f (n)  an ,
and to denote the sequence by an
Examples
n 1
3 4 5
an 
; first four terms: 2, , , ,
n
2 3 4
n 1
1 2 3 4
an  2 ; first four terms: , , ,
n
2 5 10 17
an  (1) ; first four terms:  1, 1,  1, 1
n
an  (1) n ; first four terms:  1, 4,  9, 16
n
2
Recursion formulas
A recursion formula or recurrence relation
gives ak+1 in terms of one or more of the
terms am that precede ak+1.
Examples: Find the first four terms and
the nth term for the sequence specified by
(1) a1  3 and ak 1  2ak , k  1, 2,3,
(2) a1  1 and ak 1  (k  1)ak , k  1,2,3,
Solutions
(1) a2  2a1  2  3,
a3  2a2  2  2  3  22  3,
a4  2a3  2  22  3  23  3
In general, an  2n 1  3
(2) a2  2a1  2 1,
a3  3a2  3  2 1,
a4  4a3  4  3  2 1
In general, an  n !
More examples
(3) List the first six terms of the
sequence whose nth term an is the
nth prime number. Give a
“formula” for an.
(4) The first four terms of the
1 1 1
sequence an are: 1, , ,
2 3 4
What is the 5th term?
Answers
(1) 2, 3, 5, 7, 11, 13; an = ??????
(2)
an 

2
1
1 

 (n  1)( n  2)( n  3)( n  4)  
n
 48 120 
Limits of sequences
Given a sequence an. What is the behavior
of an for very large n ? That is, as n
what can you say about an ?
Examples:
n 1
(1) an 
n
(3) an  (1)n
n 1
(2) an  2
n
(4) an  (1)n n 2
Answers
(1) 1
(2) 0
(3) No limit
(4) No limit
Two special sequences
1. Arithmetic sequences: A sequence is
an arithmetic sequence (arithmetic
progression) if successive terms differ
by a constant d, called the common
difference. That is an is an arithmetic
sequence if
ak 1  ak  d for every positive integer k
Examples
Determine whether the sequence is an arithmetic sequence
(1) 2, 5, 8, 11, ,3n  1
(2) 1, 4, 9, 16,
(3) 22, 18, 14, 10, ,
Answers:
(1) Yes
(2) No
(3) Yes, assuming the pattern goes on as
indicated
(4)What is the 12th term of the arithmetic
sequence whose first three terms are:
1, 5, 9?
(5) The sequence a1 , a2 , a3 ,
What is an for all n?
is an arithmetic sequence.
Solving the recursion formula
We know ak 1  ak  d which implies ak 1  ak  d .
Thus,
a2  a1  d
a3  a2  d  a1  2d
a4  a3  d  a1  3d

an  a1  (n  1)d
This solves (5). The solution of (4) is a12  1  (11)4  45
Geometric sequences
A geometric sequence is a sequence in which
the ratio of successive terms is a nonzero
constant r. That is,
ak 1
 r which implies
ak
ak 1  r  ak
The number r is called the common ratio.
Examples
(1)The sequence 8, 4, 2, 1, …. is a geometric
sequence. Find the common ratio and give
the 5th term.
5 5 5
(2) The sequence 5,  , ,  , 
2 4
8
is a geometric sequence, find the common
ratio and give the 6th term.
(3) an geometric sequence with common
ratio r. Give a formula for an.
Answers:
1
(1) r  ;
2
1
(2) r   ;
2
1
a5 
2
5
a6  
32
(3) Let a1  a. Then
an  a1r n 1
Function defined on intervals
Let f : X Y where X is an interval or a
union of intervals and Y is the set of real
numbers.
The graph of f is the set of all points
(x,f(x)) in the coordinate plane.
The graph of f is the graph of the
equation y=f (x).
Examples
f(x) = 2x + 1
f (x) = x2 + 1
The Elementary Functions
1. The constant functions:
f ( x)  c
where c is a constant.
The graph of f is a horizontal line c units
above or below the x-axis depending on
the sign of c.
f (x) = 2
(2) The identity function and linear functions
(a) The function f (x) = x is called the
identity function. The graph is
(b) Functions of the form
f ( x)  mx  b, m  0
are called linear functions. The graph of f
is a straight line with slope m and y - intercept
b.
f ( x)  2 x  1
NONLINEAR FUNCTIONS
(3) Functions of the form
f ( x)  ax 2  bx  c
where a, b and c are constants with a  0
are called quadratic functions. The graph of
a quadratic function is a parabola.
a>0
a<0
(4) Functions of the form
f ( x)  a 3  bx 2  cx  d
where a, b, c, and d are constants with a  0
are called cubic functions. The graph of a
cubic looks like
a>0
a<0
(5) Polynomial Functions
A function of the form
f ( x)  an x  an 1 x
n
n 1
   a1 x  a0 ,
where n is a nonnegativ e integer and
a0 , a1 ,  , an are constants with an  0,
is a polynomial function.
n is the deg ree of the polynomial ; a0 , a1 , 
are the coefficien ts.
n  0 : polynomial s of degree 0; the nonzero constant
functions.
n  1 : polynomial s of degree 1; linear functions.
n  2 : polynomial s of degree 2; quadratic functions.
and so on
p( x)
A polynomial function is a rational function : p( x) 
1
but not conversely
(6) Rational functions
A rational f unction is a function of the form
p( x)
r ( x) 
q( x)
where p and q are polynomial functions.
A polynomial function is a rational function
p( x) 
p( x)
.
1
Some graphs
f ( x) 
1
x2
f ( x) 
f ( x) 
2x 1
x2 1
x
x2 1
The Elementary Functions
(7) Algebraic functions: sums, differences,
products, quotients and roots of rational
functions.
(8) The trigonometric functions.
(9) Exponential functions.
(10) Logarithm functions.