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Chapter 1: Functions of real variables
1.1 Functions
A function is a special kind of relation where each element of X is related to one and
only one element of Y.
Determine which one is a function?
a) X Y b) X Y
a p a p
b q b q
c r c r
d s d s
c) d)
X Y X Y
a p a p
b q b q
c r c r
d s d s
Examples of a function and none function by using graphs:
GRAPH A GRAPH B
1.1.1 Domain and Range of a function
If f is a function that maps elements of set X to elements of set Y, then
Set X is the domain of the function.
Set Y is the range.
All elements of set Y are the codomain.
X Y
a p
b q
c r
d s
1.1.2 Functional notation
Let X and Y be two sets.
If for each element x X , there is exactly ONE corresponding element y Y , then X
is mapped into Y by the function f.
f : x y or f ( x ) y
1.1.3 Composite functions
Composite functions are functions that arise as a result of the combination of two or more
basic functions. The composite function f g is denoted by f g x f g x . In
general, f g is not the same as g f.
[Tutorial 1]
1.4.2 Logarithmic function
The function of the form f ( x )log ax where a 0 is called logarithmic function.
The domain of log ax is
(^) and the range is (^) .
Proposition:
Let a and b be positive real numbers with a. b 1 , a. b 0 and x. y R , then
x a ax y
y log a x
a x
log
log (^) a xy log ax log ay log (^) a 1 0
x y y
x
a a a
log log log
x
x
a
a
x y ax
y log a (^) log
When a e , then
x e x y
y ln
e x
x ln
e x
x
ln
[Tutorial 3]
1.5 Hyperbolic functions
Hyperbolic function is a class of functions which is defined in terms of exponential
function. It has interesting properties in the manner that is analogous to trigonometric
functions.
For any real number x , then
2
sinh
x x e e x
^
x x x e e
ech x
sinh( )
cos
2
cosh
x x e e x
^
x x x e e
h x
cosh( )
sec
x x
x x
e e
e e
x
x x
cosh
sinh tanh
x x
x x
e e
e e
x
x x
sinh
cosh coth
