Chapter 9
Efficiency of Algorithms
9.1
Real Valued Functions
Real-Valued Functions of a Real
Variable
• Definition
– Let f be a real-valued function of a real variable.
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).
– y = f(x) ⇔ the point (x, y) lies on the graph of f.
Example
Power Functions
• Definition
– Let a be any nonnegative number. Define pa, the
power function with exponent a, as follows:
pa(x) = xa for each nonnegative real number x.
Example
• p1/2 = x1/2 point(x,x1/2)
• p0 = x0 = 1 point(x, 1)
• p2 = x2 point(x, x2)
Graphing Function on Integers
• A real-valued function may be graphed on a
set of integers.
Multiple of a Function
• Definition
– 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, 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 the domain of f
Example
Increasing & Decreasing Functions
 x, if x  0 

f (x)  x  
x, if x  0 

Increasing & Decreasing Function
• Definition
– 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 x1 and x2 in S, if x1 < x2 then f(x1) < f(x2).
We say that f is decreasing on the set S if, and only
if, for x1 and x2 in S, if x1 < x2 then f(x1) > f(x2).
Example