Relations and Functions — Class 11 Mathematics

Core Concepts

The Cartesian product A×B contains ordered pairs (a,b) where a∈A and b∈B. A relation from A to B is a subset of A×B. A function is a relation in which every input has exactly one output.

Worked Examples

Example: A={1,2}, B={3,4}. A×B={(1,3),(1,4),(2,3),(2,4)}. The relation R={(1,3),(2,4)} is a function from A to B because each element of A occurs exactly once as a first component.

Domain, range: For f={(1,5),(2,7),(3,7)}, domain={1,2,3}, range={5,7}. The range need not contain every element of the codomain.

Explained MCQs

1. A function assigns: exactly one output to each input. Multiple inputs may have the same output.
2. For f(x)=x², f(−3)=9. Squaring removes the sign.
3. The range of f(x)=x² for real x is [0,∞) because a real square cannot be negative.

Practice

1. Determine whether {(1,2),(2,3),(1,4)} is a function.
2. Find the domain and range of f(x)=2x+1 for x∈{0,1,2,3}.

Common Mistake

Having one input mapped to two different outputs violates the definition of a function; two different inputs may share one output.