Functions is a very important topic in mathematics. These are relation which give list of output corresponding to a set of inputs. While working with functions, students come across the two very useful concepts domain and range often. |

**Given below are some examples:**

**Example 1:**Find the domain value of the function given below

F(x) = $\frac{1}{x + 15}$

**Solution:**Equate the denominator to zero for the above function

x + 15 = 0

x = - 15

Set of all variable of x is the domain function f, the interval is the (- $\infty$, - 15) $\cup$ (- 15, $\infty$)

Therefore the domain value of the given function is (-$\infty$, - 15) $\cup$ (-15, $\infty$)

**Circle is the two dimensional closed geometrical figure where the distance from the centre point to the locus coming from all points is constant.**

Domain of circle: All the feasible x values are considered to be domain of circle.

Range on the circle: All the feasible y values are considered to be range of circle.

The standard way of the circle is actually

(x - a)$^2 $ + (y - b)$^2 $ = r$^2 $

Here r is the radius of circle

(a, b) is the centre of the circle.

**Given below is an example:**

**Example 1:**Find the domain and range of the circle : x$^{2}$ + y$^{2}$ = 49

**Solution:**

The general form of a circle is

(x - a)$^{2}$ + (y - b)$^{2}$ = r$^{2}$

Now,

x$^{2}$ + y$^{2}$ = 49

$\rightarrow$ (x - 0)$^{2}$ + (y - 0)$^{2}$ = 7$^{2}$

so that x$^{2}$ + y$^{2}$ = 49 is a circle with its center at the origin and radius 7.

If you construct that circle, it's easy to see that x runs from - 7 to + 7 and y runs from - 7 to + 7.

Therefore,

DOMAIN = {x│ - 7 $\leq$ x $\leq$ 7 }

RANGE = {y│ -7 $\leq$ y $\leq$ 7 }

**Exponential Functions:**In general, the domain of the function y = a$^{x}$ is the possible x values and the range is the possible y values of that function.

For a function y = a$^{x}$ , where a > 1 (or) 0 < a < 1,

The domain consists of all the real numbers which can be written in the interval form as $( -\infty , \infty)$.

The Range consists of all positive real numbers $(0 , \infty)$. We can also observe that $\lim_{x \to \infty} a^{x} = \infty$

And, for no value of x, a$^{x}$ is 0. The graph will never meet the x-axis. It is likely to meet the x - axis at a very small value of x when a > 1, and for a very large value of x, when a< 1. Hence, for the above function, the horizontal asymptote is x-axis.

*Logarithmic Functions:*Log is the inverse function of exponentiation

Consider a function

y=log(x−3)

We are trying to find the Domain and the Range of this function, recalling that:

Domain: Includes all values of x for which the function is defined.

Range: Includes all values y for which there is some x such that y=log(x−3).

It makes no sense to write y = log(a) when a $\leq$ 0 because log(a) is defined only for positive a. So in this problem, y = log(x - 3), is defined if and only if x − 3 > 0 $\rightarrow$ x > 3, and that gives domain x $\in$ (3, $\infty$).

Range of y is all of R.

Therefore the domain is $\in$ (3, $\infty$).

Range : y $\in$ R