Page 164 - DMTH201_Basic Mathematics-1

Unit 6: Functions

If f(x ) > f(x ) when x < x , then f(x) is increasing. Notes
2 1 1 2
If f(x ) < f(x ) when x < x , then f(x) is decreasing.
2 1 1 2
If f(x ) = f(x ) for all values of x and x in I, then f(x) is constant.
2 1 1 2
If, however, the strict inequality holds in the above statements, then f(x) is strictly increasing
(or decreasing) function.
2. Monotonic Function
A function y = f(x) is said to be monotonic if y is either increasing or decreasing over its
domain, as x increases.
If the function is increasing (decreasing) over its domain, it is called monotonically increasing
(decreasing) function.
A monotonic function is also termed as a one to one function.
3. Implicit and Explicit Function
When a relationship between x and y is written as y = f(x), it is said to be an explicit
function. If the same relation is written as F(x, y) = 0, it is said to be an implicit function.
Production possibility function or the transformation function is often expressed as an
implicit function.
4. Inverse Function
If a function y = f(x) is such that for each element of the range we can associate a unique
element of the domain (i.e. one to one function), then the inverse of the function, denoted
–1
as x = f (y) g(y), is obtained by solving y = f(x) for x in terms of y. The functions f(x) and g(y)
are said to be inverse of each other and can be written as either g[f(x)] = x or f[g(y)] = y. We
note here that an implicit function F(x, y) = 0, can be expressed as two explicit functions
that are inverse of each other.
5. Symmetry of a Function
Symmetry of a function is often helpful in sketching its graph. Following types of symmetry
are often useful:
(i) Symmetry about y-axis
A function y = f(x) is said to be symmetric about y-axis if f(–x) = f(x) for all x in its
domain. For example, the function y = x is symmetric about y-axis. Such a function
2
is also known as even function.
Similarly, if g(y) = g(–y), then the function x = g(y) is said to be symmetric about x-
axis.
(ii) Symmetry about the line x = h

A function y = f(x) is said to be symmetric about the line x = h if f(h – k) = f(h + k) for
all real value k.
(iii) Symmetry about origin

A function y = f(x) is said to be symmetric about origin if f(–x) = –f(x), for all values
2
of x in its domain. For example, the function y = x is symmetric about origin. Such
a function is also known as odd function.
(iv) Symmetry about the line y = x (45° line)
Two functions are said to be symmetrical about the line y = x (45° line), if the
interchange of x and y in one function gives the other function. This type of symmetry

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