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1. 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. y2 = 4ax ; x2 + y2 = a2 ; y (y-2) (y +2) = x2

2. The mapping of function f: x ® y is said to be Many-One if two or more different elements in x have same f-image in y.

3. The mapping or function f is said to be One-One if different element in x have different f-images in y, i.e. x1 ¹ x2 Þ f (x1) ¹ f ( x2) or f (x1) = f (x2) Þ x1 = x2.

One-One mappings are also called injection or injective mappings.

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

5. The mapping 'f ' is said to be 'onto' if every element in y is the f-image 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.

One-One 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. Your browser does not support the IFRAME tag.
6. Odd and Even functions : If f(-x) = f(x), f(x) is called an even function.

e.g. f(x) = ax4+ bx2 + c, f (x) = cos x etc.

If f (-x) = -f (x), f (x) is called an odd function.

e.g. f(x) = ax3 + bx , f(x) = tan x etc.

Note that any function can be expressed as the sum of an even and odd function.

viz. 7. 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. xn , 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. x2y2 + 4xy + 3y + 5x + 6 = 0. Here y is implicit function of 'x'.

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