# Types of Functions

## Types of Functions:

(1) One-to-one Functions- A function, f: A â†’ B, is said to be one-to-one or injective if for distinct elements x1, x2 âˆˆ A, there exist f(x1), f(x2) respectively in B such that for x1 â‰  x2, f( x1) â‰  f(x2). Linear functions like f(x) = 2x – 1, f(x) = x + 2, f(x) = (2x + 1)/3 etc., are all one-to-one functions. The graphs of one-to-one functions denote straight lines. The mapping has been shown in the figure below.

(2) Many-to-one Functions- Functions that are not one-to-one are said to be many-to-one functions. Functions like y – x2, y = x3, y = | x | are many-to-one functions. The mapping has been shown in the figure below. The codomain of the function, in this case, is set B but the range is {y1, y2, y3}.

(3) Onto-(Surjective) Functions- A function, f: A â†’ B, is said to be an onto or surjective function if each element y of B has a pre-image x in A. The codomain in this case is called the Range of f. Set B in this case must also be exhausted as shown in the figure below.

(4) Into Functions- A function, f: A â†’ B, is said to be an into-function if there exists at least one element in B having no pre-image in A. In the figure below, there are two elements y4 and y5 in set B for which there is no pre-image in A. The function in the case is an into-function.

(5) Bijection- A function, f: A â†’ B, is a bijection if-

• It is one-one i.e. f(x1) = f(x2) â‡’ x1 = x2 for all x1, x2 âˆˆ A.
• It is onto i.e. for all y âˆˆ B, there exists x âˆˆ A such f(x) = y.

(6) Identity Function- Let A be a non-void set. A function, f: A â†’ A, is said to be the identity function on set A if f associates every element of set A to the element itself. Identity function is given by f(x) = x.

(7) Constant Function- A function, f: A â†’ B, is a constant function if every member of A has the same image in B under the function f. Thus, in this case f(x) = c, for all x âˆˆ A, c âˆˆ B.

(8) Exponential Function- The function is given by f(x) = ex. It lies completely above the X-axis and from left to right it rises exponentially.

(9) Logarithmic Function- It is given by f(x) = loge x. The graph of the function lies completely on the right side of the Y-axis and it rises above from left to right as shown in figure (a).

When f(x) = ex and f(x) = loge x are drawn on the same scale, it would be clear that one is reflection of the others in the line y = x (the identity function) as shown in figure (b).

(10) The Greatest Integer Function- It is defined as f(x) = [x], where [x] refers to the greatest integer contained in x. It is sometimes referred to as the Integral value function or the Staircase function (as the graph of the function looks like a staircase). The graph of the function consists of many broken pieces, each being coincident with the graph of a constant function as shown in the figure below.

(11) Trigonometric Functions- The graphs of f(x) = sin x, f(x) = cos x and f(x) = tan x are shown below.

(12) Signum Function- The function is defined as f(x) = | x |/x, when x â‰  0 and f(x) = 0, when x = 0. Clearly, (0, 0) is a point on the graph and | x |/x = 1, when x > 0, | x |/x = -1, when x < 0 are the two branches of the graph as shown in the figure below.

(13) Reciprocal Function- The function is given by f(x) = 1/x, x â‰  0. It is symmetrical about the origin as shown in the figure below.

(14) Absolute Value Function-

It is also known as the Modulus function. The graph is composed of two rays originating from the same point and is symmetrical with respect to the Y-axis. It lies completely above the X-axis.