## Types of Functions:

(1) ** One-to-one Functions-** A function, f: A

**→**B, is said to be one-to-one or injective if for distinct elements x

_{1}, x

_{2}∈ A, there exist f(x

_{1}), f(x

_{2}) respectively in B such that for x

_{1}≠ x

_{2}, f( x

_{1}) ≠ f(x

_{2}). 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 {y

_{1}, y

_{2}, y

_{3}}.

(3) ** Onto-(Surjective) Functions-** A function, f: A

**→**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.

Thus, Range of f = f(A) = {f(x): x ∈ A} Obviously, f(A) ⊆ B |

(4) ** Into Functions-** A function, f: A

**→**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 y

_{4}and y

_{5}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

**→**B, is a bijection if-

- It is one-one i.e. f(x
_{1}) = f(x_{2}) ⇒ x_{1}= x_{2}for all x_{1}, x_{2}∈ A. - It is onto i.e. for all y ∈ B, there exists x ∈ A such f(x) = y.

(6) ** Identity Function-** Let A be a non-void set. A function, f: A

**→**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

**→**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 ∈ A, c ∈ B.

(8) ** Exponential Function-** The function is given by f(x) = e

^{x}. It lies completely above the X-axis and from left to right it rises exponentially.

(9) ** Logarithmic Function-** It is given by f(x) = log

_{e}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) = e^{x }and f(x) = log_{e} 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

**or the**

*Integral value 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.**

*Staircase function*(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 ≠ 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 ≠ 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.

Find whether the function, f: I Example-→ I defined by f(x) = x^{2} + 5, for all x ∈ I is one-one or not. Let xSolution-_{1}, x_{2} ∈ I (domain) such that f(x_{1}) = f(x_{2})⇒ x _{1}^{2} + 5 = x_{2}^{2} + 5⇒ x _{1}^{2} = x_{2}^{2}⇒ x _{1} = ± x_{2}⇒ x _{1} ≠ x_{2}∴ f is not one-one. |

State whether the following functions are onto-functions:Example- (i) f: R → R given by f(x) = x^{2} + 3 for all x ∈ R Let y ∈ R (co-domain) such that f(x) = ySolution-⇒ x ^{2} + 3 = y⇒ x ^{2} = y – 3⇒ x = √(y – 3) For all values of y < 3, x ∉ R (domain). ∴ the function is into. (ii) f: I → I defined by f(x) = 5x + 2 for all x ∈ I Let y ∈ I (co-domain) such that f(x) = ySolution-⇒ 5x + 2 = y ⇒ 5x = y -2 ⇒ x = (y – 2)/5 For some values of y, x ∉ I. ∴ f is an into function. (iii) f: Q → Q given by f(x) = 2x – 5 for all x ∈ Q Let y ∈ Q (co-domain) such that f(x) = ySolution-⇒ 2x – 5 = y ⇒ 2x = y + 5 ⇒ x = (y + 5)/2 For each y ∈ Q (co-domain), x ∈ Q (domain). ∴ f is an onto function. |

Show that g: R Example-→ R defined by g(x) = 2x^{3} + 8 for all x ∈ R is a bijection.Solution-g is one-one:Let x _{1}, x_{2} ∈ R (Domain) such that g(x_{1}) = g(x_{2})⇒ 2x _{1}^{3} + 8 = 2x_{2}^{3} + 8⇒ 2x _{1}^{3} = 2x_{2}^{3}⇒ x _{1}^{3} = x_{2}^{3}⇒ x _{1} = x_{2}∴ g is one-one. g is onto:Let y ∈ R (co-domain) such that g(x) = y ⇒ 2x ^{3} + 8 = y⇒ 2x ^{3} = y -8⇒ x ^{3} = (y -8)/2⇒ x = [(y -8)/2] ^{1/3}For each y ∈ R (co-domain), x ∈ R (domain). ∴ g is onto. Thus, g is a bijection. |

Show that the function, g: A Example-→ B defined by g(x) = (x – 3)/(x – 4) where A = {x: x ∈ R, x ≠ 3} and B = {y: y ∈ R, y ≠ 1}, is a bijection.Solution-g is one-one:Let x _{1}, x_{2} ∈ A (Domain) such that g(x_{1}) = g(x_{2})⇒ (x _{1} – 3)/(x_{1} – 4) = (x_{2} – 3)/(x_{2} – 4)⇒ x _{1}x_{2} – 4x_{1} – 3x_{2} + 12 = x_{1}x_{2} – 3x_{1} – 4x_{2} + 12⇒ – 4x _{1} – 3x_{2} = – 3x_{1} – 4x_{2}⇒ -x _{1} = -x_{2}⇒ x _{1} = x_{2}∴ g is one-one. g is onto:Let y ∈ B (co-domain) such that g(x) = y ⇒ (x – 3)/(x – 4) = y Apply Componendo and Dividendo: ⇒ 2x – 7 = (y + 1)/(y – 1) ⇒ 2x = (y + 1)/(y – 1) + 7 ⇒ 2x = (y + 1 + 7y -7)/(y – 1) ⇒ 2x = (8y – 6)/(y – 1) ⇒ x = (4y – 3)/(y – 1) For each y ∈ B = R – {1}, x ∈ A = R – {4} ∴ g is onto. Thus, g is a bijection. |