Spring 2016
COMP 2300
Discrete Structures for Computation
Chapter 11.1
Real-Valued Functions of a Real Variable and
Their Graphs
Donghyun (David) Kim
Department of Mathematics and Physics
North Carolina Central University
1
Cartesian Plane
• A Cartesian plane or two-dimensional Cartesian
coordinate system is a pictorial representation of R  R
obtained by setting up a one-to-one correspondence
between ordered pairs of real numbers and points in a
Euclidean plane.
Horizontal axis
(0 ,3 )
(2 ,1)
( 2 ,2 .5 )
Vertical axis
Origin: the intersection of the two axes
Spring 2016 COMP 2300
Donghyun (David) Kim
2
Department of Mathematics and Physics
North Carolina Central University
A Graph of A Function
• A real-valued function of a real variable is a function
from one set of real numbers to another.
• If f is such a function, then for each real number x in
the domain of f, there is a unique corresponding real
number f(x).
• The graph of f is the set of all points (x, y) in the
Cartesian coordinate plane with the property that x is
in the domain of f and y=f(x).
Spring 2016 COMP 2300
Donghyun (David) Kim
3
Department of Mathematics and Physics
North Carolina Central University
A Graph of A Function – cont’
y=f(x)  the point (x, y) lies on the graph of f.
f (x)
( x, f ( x))
Graph of f
x
4
Spring 2016 COMP 2300
Donghyun (David) Kim
Department of Mathematics and Physics
North Carolina Central University
Power Functions
• Let a be any nonnegative real number. Define pa , the
power function with exponent a, as follows:
pa ( x)  x a for each nonnegative real number x.
5
Spring 2016 COMP 2300
Donghyun (David) Kim
Department of Mathematics and Physics
North Carolina Central University
Floor Functions
• The floor of a number is the integer immediately to its
left on the number line. More formally, the floor
function F is defined by the rule
F ( x)  x  = the greatest integer that is less than or equal to x
= the unique integer n such that n  x  n 1 .
6
Spring 2016 COMP 2300
Donghyun (David) Kim
Department of Mathematics and Physics
North Carolina Central University
Graphing Functions Defined on
Sets of Integers
• Many real-valued functions used in computer science
are defined on sets of integers and not on intervals of
real numbers.
7
Spring 2016 COMP 2300
Donghyun (David) Kim
Department of Mathematics and Physics
North Carolina Central University
Graph of Multiple of a Function
• Let f be a real-valued function of a real variable and let
M be any real number. The function Mf, called the
multiple of f by M or M times f, is the real-valued
function with the same domain as f that is defined by
the rule ( Mf )( x)  M  ( f ( x)) for all x in domain of f.
yx
y  2x
8
Spring 2016 COMP 2300
Donghyun (David) Kim
Department of Mathematics and Physics
North Carolina Central University
Increasing and Decreasing
Functions
• Consider the absolute value function, A, which is
defined as follows:
 x ifx  0
A( x) | x | 
 x ifx  0
for all real numbers x.
y | x |
9
Spring 2016 COMP 2300
Donghyun (David) Kim
Department of Mathematics and Physics
North Carolina Central University
Increasing and Decreasing
Functions – cont’
• Let f be a real-valued function defined on a set of real
numbers, and suppose the domain of f contains a set S. We
say that f is increasing on the set S if, and only if,
for all real numbers x1 and x2 in S, if x1  x2 then f ( x1 )  f ( x2 ).
• We say that if is decreasing on the set S if, and only if,
for all real numbers x1 and x2 in S, if x1  x2 then f ( x1 )  f ( x2 ).
• We say that f is an increasing (or decreasing) function if, and
only if, f is increasing (or decreasing) on its entire domain. 10
Spring 2016 COMP 2300
Donghyun (David) Kim
Department of Mathematics and Physics
North Carolina Central University