Q1. Find the domain and range of f(x) = 1/(x − 2).
f is undefined when the denominator is 0, i.e. x = 2
Domain = all real numbers except 2, i.e. ℝ − {2}
Range = all real numbers except 0, i.e. ℝ − {0} (f(x) never equals 0)
Relations and Functions — Class 11
A function is a rule pairing each input with exactly one output. Learn to find the domain and range and spot the standard function types.
Cartesian productA × B = the set of all ordered pairs (a, b) with a ∈ A and b ∈ B. If |A| = m, |B| = n, then |A×B| = mn.
RelationAny subset of A × B — a set of ordered pairs linking elements of A to B.
FunctionA relation where every input (domain element) has EXACTLY ONE output (no input repeats with a different output).
Domain, codomain, rangeDomain = all valid inputs; codomain = the set outputs are drawn from; range = the outputs actually achieved.
Solved examples
Q2. If A = {1, 2} and B = {3, 4}, find A × B.
Pair every element of A with every element of B
A × B = {(1,3), (1,4), (2,3), (2,4)}
Formula sheet
| Quantity | Formula |
|---|---|
| Size of Cartesian product | |A×B| = |A| × |B| |
| Number of relations | 2^(mn), where |A|=m, |B|=n |
| Modulus function | |x| = x if x≥0, −x if x<0 |
Common mistakes & exam wins
- Every function is a relation, but not every relation is a function — check that no input repeats with a different output.
- For a fraction, exclude values that make the denominator zero from the domain.
- For a square root, the domain requires the expression inside to be ≥ 0.
Frequently asked questions
What is the difference between a relation and a function?
A relation is any set of ordered pairs; a function is a relation where each input has exactly one output.
What is the domain of a function?
The complete set of valid input (x) values for which the function is defined.
How many elements does A × B have?
If A has m elements and B has n elements, A × B has mn elements.