Download Functions

Topics
• Intro. to Functions (7.1)
• More functions (9.1)
• Efficiency of algorithms/functions (9.2)
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Original author of the slides:
Vadim Bulitko
University of Alberta
http://www.cs.ualberta.ca/~bulitko/W04
Modified by T. Andrew Yang
([email protected])
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Definition
• Intuition: a function takes input and produces one
output:
– f(x) = x2
– f(x) = sin(x)
• Formalism:
– Domain type:
Df
– Range type:
Rf
– [Mapping] Graph:
• Gf = { <x, f(x)> | xDf, f(x)Rf}  Df x Rf
– For every xDf there is at most one pair <x,f(x)>  Gf
• Graphs of sample functions:
– Let D = {1,2,3,4,5}. f(x) = x2, x  D.
– f(x) = 1/x, x  R.
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Example: f(x) = x2
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Functional Property
For every x there is at
most one y such that
y=f(x) [y=1/x]
There is an x such that more
than one y satisfy y=f(x)
[x2+y2=25]
Example: x=0, y1=5, y2=-5
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Domain & Range
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Questions?
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Graphs of Real-Valued Functions
of a Real Variable
• A real-valued function of a real variable is a function
from one set of real numbers to another.
• 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).
• Exercises: Show the graphs of the following functions.
–
–
–
–
–
f(x) = a, where a is a number.
f(x) = ax, where a is a number.
f(x) = x2
f(x) = x1/2
f(x) = log bx, where b is a number.
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Multiple of f by M
• Let f be a real-valued function of a real variable, and let
M be any real number.
• M f, called the multiple of f by M (or M times f), is the
real-valued function with the same domain as f and
(M f) (x) = M f(x), for all x in the domain of f.
• See Example 9.1.4 (p.514-515)
• Show the graphs of the following functions:
f(x) = x
f(x) = 2x
f(x) = x2
f(x) = 3x2
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Tool: http://www.lukewallin.co.uk/graph/newsuite.htm
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Increasing vs Decreasing
functions
• Let f be a real-valued function defined on a set of real
numbers, and suppose the domain of f contains a set
S.
• f is increasing on the set S iff
for all x1 and x2 in S, x1 < x2  f(x1) < f(x2).
• f is decreasing on the set S iff
for all x1 and x2 in S, x1 < x2  f(x1) > f(x2).
• See Figure 9.1.5 (p.516) for examples of increasing
and decreasing functions.
• Examples of increasing or decreasing functions?
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Exercises
p.517:
2, 3, 4,
5, 6, 7,
14, 15, 16
21, 22
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Questions?
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