
Many valued function: To each value of x, suppose
there are more than one value of y, then y is called a multiple
valued or many valued function.
e.g. y^{2} = 4ax ; x^{2}
+ y^{2} = a^{2} ; y (y2) (y +2) = x2

The mapping of function f: x ®
y is said to be ManyOne if two or more different elements
in x have same fimage in y.

The mapping or function f is said to be OneOne
if different element in x have different fimages in y, i.e.
x_{1} ¹ x_{2} Þ
f (x_{1}) ¹ f ( x_{2})
or f (x_{1}) = f (x_{2}) Þ
x_{1} = x_{2}.
OneOne mappings are also called injection
or injective mappings.

The mapping 'f ' is said to be 'into' if there
exists at least one element in y which is not the fimage of
any element in x. Note that in this case range of ’f’ is the
proper subset of y.

The mapping 'f ' is said to be 'onto' if every
element in y is the fimage of at least one element in x. In
this case, the range of 'f ' is equal to y. 'Onto' mapping is
also called surjection or Surjective mappings.
OneOne and onto mappings are called bijection
or bijective mappings.
If the domain and codomain of a function f
are both the same, then f is called an operator or
transformation on x.

Odd and Even functions : If f(x) = f(x), f(x)
is called an even function.
e.g. f(x) = ax^{4}+ bx^{2}
+ c, f (x) = cos x etc.
If f (x) = f (x), f (x) is called an
odd function.
e.g. f(x) = ax^{3} + bx , f(x) = tan x etc.
Note that any function can be expressed as the sum of an
even and odd function.
viz.

Explicit and Implicit functions : A function
is said to be explicit when expressed directly in terms of the
independent variable or variables. e.g. y = e^{x}.
x^{n} , y = r sin q etc.
But if the function cannot be expressed
directly in terms of the independent variable (or variables),
the function is said to be implicit. e.g. x^{2}y^{2}
+ 4xy + 3y + 5x + 6 = 0. Here y is implicit function of 'x'.