# Function

A function is a rule which associates two or more than two variables. Functions can be understood with the help of some very basic examples:

- Area of a circle (A) = πr
^{2}, where r is the radius.*f*(*r*) - If the distance (d) is constant, then the time taken (t) to cover that distance (d) will be dependent on the value of the speed (v). This can be written mathematically as
*t*=*f*(*v*)*y*=*f*(*x*)

**Domain and range of**In case of

*y*=*f*(*x*)*y*=

*f*(

*x*), values of

*y*are dependent upon the corresponding values of

*x*. Here

*y*is known as the dependent variable and

*x*is known as the independent variable.

All the real values of

*x*for which atleast one real value of*y*exists, are known as the domain of this function*y*=*f*(*x*)All the possible real values of

*y*are known as the range of this function*y*=*f*(*x*).While finding out the domain, the variable generally takes a range of values unlike the case of equations where the variable will, mostly, take one value or a discrete set of values in interval notation. Hence, it is important to understand the standard notations which are used to represent the solution set of the variable as well as the interval notation.

- (
*a, b*) read as “open interval*a, b*” means all real numbers between a and b excluding*a*and*b*;*a*<*b*. - [
*a*,*b*] read as “closed interval*a*,*b*” means all real numbers between*a*and*b*including*a*and*b*;*a*<*b.* - [
*a*,*b*) means all numbers between*a*and*b*, with*a*being included and*b*excluded;*a*<*b*. - (
*a*,*b*] means all numbers between*a*and*b*, with*a*being excluded and*b*included;*a*<*b*.

Example-1

Represent all real numbers between 1 and 10 in the interval from where

- 1 and 10 are included,
- 1 and 10 are excluded
- 1 is included and 10 is excluded and
- 1 is excluded and 10 is included

Solution

- [1, 10] i.e., 1 ≤
*x*≤ 10 and*x*is a real number. - (1, 10) i.e., 1 <
*x*< 10 and*x*is a real number. - [1, 10) i.e., 1 ≤
*x*< 10 and*x*is a real number. - [1, 7] i.e., 1 ≤
*x*≤ 7 and*x*is a real number.

Process to find out the domain of *y = f(x)*

- Denominator should not be equal to zero.
- Expression under square root should be a non-negative.

Example-2

Find the domain of the definition of function

*y*=*f*(*x*) = .Solution

We know that the expression under square root should be non-negative.

So, 4 –

*x*≥ 0, or,*x*≤ 4And

*x*– 6 ≥ 0, or,*x*≥ 6.Arranging these values on the numberline:

*x*is common. Hence the domain of

*y*=

*f*(

*x*) is not defined.

Example-3

The domain of

*y*=*f*(*x*) = isSolution

For
It is true only if

*y*=*f*(*x*) to be defined, |*x*| −*x*>0, or, |*x*|>*x*.*x*< 0. So, the domain is −∝ <*x*< 0.Example-4

Are the following functions identical?

*y*=*f*(*x*) =*x**y*=*h*(*x*) =Solution

Domain of
Domain of
Because the domains are not the same for

*f*(*x*) = all the real values*h*(*x*) = all the real values except at*x*= 0.*f*(*x*) and*h*(*x*), so*f*(*x*) and*h*(*x*) are not identical.