(Last Updated On: June 25, 2023)

Based on the likelihood of the given function being defined in the real set, the domain and range of a function may be determined. Let’s take a closer look at domain and range, which are discussed in depth here.

Domain

The domain of a function is defined as the full set of potential values for independent variables, or as the set of all potential values that qualify as inputs to a function. The domain is — the denominator of the fraction is not zero, and the digit under the square root bracket is positive. (If the function has a fractional value.)

How to Find the Domain of a Function

  • To discover the domain, we must examine the values of the independent variables, which are permitted to be used as previously stated, i.e. no zero at the bottom of the fraction and no negative sign within the square root.
  • In general, the domain of a function is defined as the set of all real numbers (R), subject to various constraints. They are as follows:
    The domain is “the set of all real numbers” when the provided function is of the kind f(x) = 2x + 5 or f(x) = x2 – 2.
    The domain is the set of all real numbers except one when the supplied function is of the type f(x) = 1/(x – 1).
  • The interval may be supplied together with the function in some circumstances, such as f(x) = 3x + 4, 2 x 12. x may take any number of numbers between 2 and 12 as input (i.e. domain).
  • Domain limitations relate to the ranges of values for which a function cannot be defined.

Range

The Range of a function is the set of all the function’s outputs, or, after replacing the domain, the whole set of all potential values as dependent variable outcomes.

How to Find the Range of a Function

Take the function y = f(x).

  • The range of the function is the distribution of all the y values from least to the maximum.
  • Substitute all the values of x in the supplied expression of y to see if it is positive, negative, or equal to other values.
  • Determine y’s minimum and maximum values.
  • Then create a graph for it.

“It is possible to restrict the range (i.e. the output of a function) by redefining the codomain of that function,” says one intriguing aspect concerning range and codomain.

Example: Find the domain and range of a function f(x) = (2x – 1)/(x + 4).
Solution:
                   Given function is:

                   f(x) = (2x – 1)/(x + 4)

                   We know that the domain of a function is the set of input values for f, in which the function is real and defined.

                  The given function is not defined when x + 4 = 0, i.e. x = -4

                  So, the domain of given function is the set of all real number except -4.

                  i.e. Domain = (-∞, -4) U (-4, ∞)

                  Also, the range of a function comprises the set of values of a dependent variable for which the given function is defined.

                  Let y = (2x – 1)/(x + 4)

                  xy + 4y = 2x – 1

                 2x – xy = 4y + 1

                 x(2 – y) = 4y + 1

                 x = (4y + 1)/(2 – y)

                This is defined only when y is not equal to 2.

                Hence, the range of the given function is (-∞, 2) U (2, ∞).

Cuemath

Cuemath is a math education curriculum for students in kindergarten to grade 12. This curriculum is available in India in home-based zones. Math is a life skill that cannot be taught idly, as per Cuemath primary principle. Students should take an active role in class and solve tasks that become more difficult as time goes on. Problem sets are given to students by their Cuemath teachers, and they must work their way through them. In the home-based centers, problem sets are delivered as workbooks. A teacher is available to explain the fundamentals and dispel any concerns. The Cuemath website is day to day updated by the teacher for a student’s ongoing active participation and solving all worksheets and puzzles.