2.2 Functions  (Page 3/3)

Example

Problem 1 : Let “A” be the set of first three natural numbers. A real function is defined as :

$$f:A\to Nbyf\left(x\right)=x^2+1$$

Find (i) domain of “f” (ii) range of “f” (iii) co-domain of “f” (iv) f(3) and (v) pre-images of 2 and 4.

Solution : Here set “A” is the domain of “f”. Hence,

$$\text{Domain of "f"}=A=\left\{\mathrm{1,2,3}\right\}$$

For determining the range, we need to find images for the each element of the domain as :

$$\mathrm{For}x=\mathrm{1,}f\left(x\right)=x^2+1=1^2+1=2$$

$$\mathrm{For}x=\mathrm{2,}f\left(x\right)=x^2+1=2^2+1=5$$

$$\mathrm{For}x=\mathrm{3,}f\left(x\right)=x^2+1=3^2+1=10$$

Hence, range of function is given as :

$$\text{Range of "f"}=A=\left\{\mathrm{2,5,10}\right\}$$

Co-domain of the function is the second set on which the elements of first set are mapped. It is given that “f” is a function from set “A” to set “N”. Hence,

$$\text{Co-domain of "f"}=N$$

The image of set for x = 3 has been already calculated. It is :

$$\Rightarrow f\left(3\right)=x^2+1=3^2+1=10$$

For pre-image of f(x) = 2, we have :

$$\Rightarrow f\left(x\right)=2=x^2+1$$

$$\Rightarrow x^2=1$$

$$\Rightarrow x=\mathrm{1,}-1$$

But, only "1" is an element of domain set "A" - not "-1". Hence, "pre-image" of "2" is "1".

For pre-image of f(x) = 4, we have :

$$\Rightarrow f\left(x\right)=4=x^2+1$$

$$\Rightarrow x^2=3$$

$$x=\sqrt{\mathrm{3,}}-\sqrt{3}$$

But it is given that domain is first three natural numbers only. Thus, we conclude that “4” has no pre-image.

Numbers of functions

We can find out maximum numbers or total possible numbers of functions that can be generated by the rule from given domain and co-domain sets, provided these sets are finite sets. We have noted that the total numbers of relations generated from Cartesian product of two sets “A” and “B” is given by :

$$N=2^{pq}$$

where “p” and “q” are the finite numbers of elements in sets “A” and “B”.

However, function is a special relation, in which each element of set ”A” is related to exactly one element of set “B” - unlike in the case of power set in which we count all possible combinations. Hence, number of possible relations is not same as the numbers of possible functions.

For determining total numbers of functions from two given sets, let us consider that “m” and “n” denote the numbers of elements in domain “A” and co-domain “B” respectively. Then, an element of domain can combine with one of the “n” elements in “B”. Hence, total numbers of such relations for a total of “m” elements in set “A” is :

$$N_f=mXn=mn$$

Finite and infinite functions

The numbers of elements of ordered pair in the function set is equal to the numbers of elements in domain set. This follows from the fact that every element of domain set “A” is related to an unique element in “B”. Thus, if domain is a finite set, then the resulting function is finite. Consider the earlier example, when A = {1,2,3} and function is defined as :

$$f:A\to Nbyf\left(x\right)=x^2+1$$

The function set is a finite set :

$$f=\{\left(\mathrm{1,2}\right),\left(\mathrm{2,5}\right),\left(\mathrm{3,10}\right)\}$$

On the other hand, if we expand this function by defining the relation from the infinite set of natural numbers, “N” to “N”, then resulting set of ordered pair is an infinite set and so is the function :

$$g:N\to Nbyf\left(x\right)=x^2+1$$

The resulting function set, in the set builder form, is given as :

$$g=\{\left(x,y\right):y=x^2+\mathrm{1,}wherex,y\in N\}$$

Function graphs

Here, we shall introduce an alternative way to represent a function. We should be aware that we can define a function even with a graph. Graphical representation of function is intuitive and revealing about their characteristics.

Function is a set of ordered pairs between “x” and “y” from domain and co-domain sets respectively in accordance with certain rule. If we look closely at the function set, then it is easy to realize that the elements of ordered pair (x,y) can be considered to be coordinates “x” and “y” of a planar coordinate system.

We represent independent variable, “x” i.e. the element of domain set “A” as abscissa along x-axis and dependent variable, “y”, i.e. the element of co-domain “B” as ordinate along y-axis. A point on the plot represented by coordinate (x,y) is an instance of or value of the function for a given value of “x”. Compositely, the graph itself is the collection of all such points, which form part of the function set.

For example, we draw a graph, which is defined as :

$$f:N\to Nbyf\left(x\right)=x,wherex\in N$$

In order to plot the function, we evaluate function values for values of “x” :

$$Forx=\mathrm{1,}y=1$$

$$Forx=\mathrm{2,}y=2$$

$$\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots$$

$$\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots$$

$$Forx=n,y=n$$

Note that plot of the function is a collection of discrete points only.

For the plot to be continuous, it is clear that the domain and co-domain of the function should be set of real numbers. In that case, we can define the function as :

$$g:R\to Rbyf\left(x\right)=x,wherex\in R$$

The corresponding plot is a bisector straight line, passing through the origin, as shown in the figure here :

Classification of real functions

Real functions can be classified from different point of views. Here, we present few major classifications.

Based on expression types

1: Algebraic function : The function (function rule) consists of algebraic expression, consisting of terms, which are constituted of constants and variables. The variables of algebraic expressions may be raised to a constant exponent. Example :

$$\frac{\sqrt{\left(3x^2+2\right)}}{x};x\ne 0$$

2: Transcendental function : The non algebraic functions are called transcendental functions. They include logarithmic, exponential, trigonometric and inverse trigonometric functions etc. Example :

$$\sin x+\cos x$$

Based on independent and dependent variables

1: Explicit function : A function is an explicit function, if its dependent variable can be expressed in terms of independent variables only. Example :

$$y=x^2+1$$

2: Implicit function : A function is an implicit function, if its dependent variable can not be expressed in terms of independent variables only. Example :

$$xy=\sin \left(x+y\right)$$