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# Domain and Range

Top
 Sub Topics 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. Domain of the function means the group coming from all x values or even all inputs. These input values when applied to the function, must produce corresponding output values. Domain of the function is the set of all the so-called values or volumes that go inside function to present outputs. The set of all the output values is termed as a range. We can say how the range of a function will be the set of many possible output beliefs corresponding to permissible input values. The range is usually calculated as hundreds of values that go toward Y axis. The number is an interval among the uppermost number for the minor number.

## Domain and Range of a Function

For any function y = f (x), the domain is the set of all the possible values of x, where x is the independent variable. Range of a function is the set of all the corresponding values of y, for every value of x,
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, $\infty$)

## Domain and Range of a Circle

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 }

## Domain and Range of Exponential and Logarithmic Functions

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