How to Find Domain and Range of Function

Here you will learn what is domain, codomain and range of function and how to find domain and range of function with example.

Let’s begin –

Let f : A \(\rightarrow\) B is a function. Then, the set A is known as the domain of f and the set B is known as the codomain of f. The set of all f-images of elements of A is known as the range of f or image set of A under f and is denoted by f(A).

Domain of f = { a | a \(\in\) A, (a, f(a)) \(\in\) f }

Range of f = { f(a) | a \(\in\) A, f(a) \(\in\) B }

Real Function : 

A function f : A \(\rightarrow\) B is called a real valued function, if B is a subset of R (set of all real numbers).

If A and B both are subsets of R, then f is called a real function.

Domain and Range of Function :

Domain of Function :

The domain of the function f(x) is the set of all those real numbers for which the expression for f(x) or the formula for f(x) assumes real values only.

Range of Function :

The range of function is the set of all the real values taken by f(x) at points in its domain. In order to find range of function,  we use following algorithm.

Algorithm :

i) Put y = f(x)

ii) Solve the equation y = f(x) for x in terms of y. Let x = \(\phi(y)\).

iii) find the values of y for which the values of x, obtained from x = \(\phi(y)\),  are real and in the domain of f.

iv) The set of all values of y obtained in step III is the range of f.

Example : Find the domain and range of function f(x) = \(x-2\over 3-x\).

Solution : we have,

f(x) = \(x-2\over 3-x\)

Domain of f : Clearly f(x) is defined for all x satisfying 3 – x \(\ne\) 0 i.e. x \(\ne\) 3

Hence, Domain of f is R – {3}.

Range of f : Let y = f(x), i.e

y = \(x-2\over 3-x\)

\(\implies\) 3y – xy = x – 2

= x(y + 1) = 3y + 2

\(\implies\) x = \(3y + 2\over y + 1\)

Clearly, x assumes real values for all y expect y + 1 = 0 i.e. y = -1

Hence, Range of f is R – {-1}.

Hope you learnt how to find domain and range of function, learn more concepts of function and practice more questions to get ahead in the competition. Good luck!

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