Syllabus Unit 3 – Calculus Functions and Their Graphs Domain, Range and Codomain of a Given Function

Domain, Range and Codomain of a Given Function

Master Domain, Range, and Codomain in CBSE Class 11 Applied Mathematics. Learn to identify function mappings with step-by-step solved examples.

Definition of Domain, Codomain, and Range

A function f: A → B is a relation where every element in set A has exactly one image in set B. The set A is called the domain, and set B is the codomain. The set of all actual images of elements of A in B is called the range.

Range(f) = {f(x) | x ∈ Domain(f)} ⊆ Codomain(f)
Example: Solved Example: Identify components of f: {1, 2, 3} → {2, 4, 6, 8} where f(x) = 2x
Show Step-by-Step Solution

Step 1: Domain is the input set {1, 2, 3}.
Step 2: Codomain is the target set {2, 4, 6, 8}.
Step 3: Calculate images: f(1)=2, f(2)=4, f(3)=6. Range is {2, 4, 6}.
Answer: Domain={1, 2, 3}, Codomain={2, 4, 6, 8}, Range={2, 4, 6}.

Finding Domain of Algebraic Functions

For real-valued functions, the domain is the set of all real numbers for which the expression is defined. We must exclude values that cause division by zero or result in the square root of a negative number.

f(x) = 1/g(x) ⟹ g(x) ≠ 0; f(x) = √g(x) ⟹ g(x) ≥ 0
Example: Solved Example: Find domain of f(x) = 1/(x - 5)
Show Step-by-Step Solution

Step 1: The denominator cannot be zero.
Step 2: Set x - 5 = 0 ⟹ x = 5.
Step 3: Exclude 5 from the set of real numbers.
Answer: Domain = R - {5}.

Determining Range of Functions

The range is the set of output values y = f(x) for all x in the domain. For simple functions, we express x in terms of y and find the values of y for which x is defined.

y = f(x) ⟹ x = f⁻¹(y)
Example: Solved Example: Find range of f(x) = 2x + 3 for x ∈ {0, 1, 2}
Show Step-by-Step Solution

Step 1: Substitute x=0: f(0)=3.
Step 2: Substitute x=1: f(1)=5.
Step 3: Substitute x=2: f(2)=7.
Answer: Range = {3, 5, 7}.