Composite function : Let f : A ® B and g : B® C be two functions and x be any element of A. Then f(x) Î B. Let y = f(x) Since B is the domain of the function g and C is its codomain, g(y) Î C. Let z = g (y) then z = g (y) = g[f(x)]Î C. This shows that every element x of set A related to unique element z = g [f(x)] of C. This gives rise to a function from the set A to the set C. This function is called the composite of f and g. It is denoted by ‘g of ’ and we have
If f : A® B and g : B® C are two functions then the composite of f and g is the function gof : A® C given by
(gof) (x) = g [ f(x) ] x Î A
A function ' f ' is such that f (x + p) = f(x) x or f(x) and f(x + p) are both undefined, then p is the period of f :
For example, ' sin e ' has period 2p rad. Since sin (x+2p) = sin x x
Linear function: f(x) = ax + b, a ¹
0 is called a linear function. The highest degree of
x is 1. It is always oneone onto.
Now we know that a linear equation in y variables
can be expressed in the form:
ax + by + c = 0, a ¹
0 , b ¹ 0. This represents a
straight line. An important part of a straight line is its slope
'm'. It is a number of units the line climbs (or falls) vertically
for each unit of horizontal change from left to right.
Thus slope (m)
=
\ m =
but Dy = y_{1}  y_{2}
and Dx = x_{1}x2
\ m =
or
Note that :

slope of a horizontal line or any line parallel
to xaxis is always 0, as vertical change remains 0 and horizontal
change remains constant.

slope of a vertical line or any line parallel
to yaxis, is said to be undefined, in
other words, it has no slope. Since horizontal change (i.e.
y) remains zero with vertical change (i.e. x) value remains
constant.

If q indicates
the inclination of an oblique line with then
slope (m) = tan q.

If q is acute,
the line has slope (m) which is positive and if q
is obtuse, line has slope (m) which is
negative.

If two straight lines are parallel we have
their slopes being equal, i.e. [ m_{1} = m_{2}
] and two straight lines are mutually perpendicular,the product
of their slopes equals to 1, i.e. [ m_{1} ·
m_{2 }= 1 ] Þ m_{1}
=

Index
Introduction
1.1 Functions And Mapping
1.2 Functions, Their Graphs and Classification
1.3 Rules for Drawing the Graph of a Curve
1.4 Classification of Functions
1.5 Standard Forms for the equation of a straight line
1.6 Circular Function and Trigonometry
Chapter 2
