Sales Toll Free No: 1-855-666-7446

Domain and Range

Top

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

Back to Top
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

Back to Top
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

Back to Top
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