Class 11 Maths · Original EZT practice
Relations and Functions (Class 11 Chapter 2): summary and practice
Ordered pairs are the raw material here: the cartesian product lists every possible pair, a relation picks some of them, and a function is the special relation that gives each input exactly one output. All questions are original EZT tutor practice following the chapter's own section order.
6 original worked examples, 6 fresh self-check questions and 4 mixed questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 21 October 2026:
1. Cartesian product of sets
A x B is the set of all ordered pairs (a, b) with a from A and b from B. If A has p elements and B has q, then A x B has p x q pairs - and order matters, so A x B and B x A differ unless A = B.
2. Relations
A relation from A to B is any subset of A x B. The set of first entries is the domain and the set of second entries actually used is the range, which sits inside the codomain B.
3. Functions
A function is a relation where every input has exactly one output. The chapter then graphs the familiar ones - identity, constant, polynomial, rational, modulus and signum - and reads domain and range off each.
4. Algebra of real functions
Real functions add, subtract, multiply and scale pointwise: (f + g)(x) = f(x) + g(x) and so on. Division f/g needs g(x) not equal to zero, which can shrink the domain.
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Count pairs in a cartesian product
Choose the method: n(A x B) = n(A) x n(B).
EZT practice question
A = {1, 2} and B = {a, b, c}. How many ordered pairs are in A x B?
Show the worked solution
- n(A) = 2 and n(B) = 3.
- n(A x B) = 2 x 3.
Answer: 6 pairs.
Check: List them: (1, a), (1, b), (1, c), (2, a), (2, b), (2, c) - all six.
Watch for this: Ordered pairs: (1, a) and (a, 1) are different, and A x B is not B x A.
Now try it yourself
A = {0, 1}, B = {1, 2} - list A x B.
Check your answer
(0, 1), (0, 2), (1, 1), (1, 2) - 4 pairs.
2 x 2 = 4 ordered pairs.
Count relations between two sets
Choose the method: A relation is any subset of A x B, so count the subsets.
EZT practice question
A = {1, 2} and B = {a, b, c}. How many relations are there from A to B?
Show the worked solution
- A x B has 6 pairs.
- Every subset of A x B is a relation, and a 6-element set has 2^6 subsets.
Answer: 2^6 = 64 relations.
Check: The count includes the empty relation and all of A x B itself.
Watch for this: Do not answer n(A) x n(B) = 6 - that counts the pairs, not the relations.
Now try it yourself
How many relations from a 2-element set to a 2-element set?
Check your answer
16.
A x B has 4 pairs, so 2^4 = 16 relations.
Count functions between two sets
Choose the method: Each input picks one of n(B) outputs.
EZT practice question
A = {1, 2} and B = {a, b, c}. How many functions are there from A to B?
Show the worked solution
- Each of the 2 inputs independently chooses one of 3 outputs.
- Total = 3 x 3.
Answer: 3^2 = 9 functions.
Check: Fewer than the 64 relations - functions are the well-behaved subset.
Watch for this: The exponent is n(A), the base is n(B) - swapping them gives 2^3 = 8, the wrong count.
Now try it yourself
How many functions from a 3-element set to a 2-element set?
Check your answer
8.
2^3 = 8.
Find the domain of a function
Choose the method: Exclude inputs that break the rule.
EZT practice question
Find the domain of f(x) = sqrt(x - 2) and of g(x) = 1/(x - 3).
Show the worked solution
- For f: the root needs x - 2 >= 0, so x >= 2.
- For g: the denominator cannot be zero, so x cannot be 3.
Answer: Domain of f is x >= 2; domain of g is all real numbers except 3.
Check: Test an edge value: f(2) = 0 works, but g(3) is undefined.
Watch for this: A square root needs its contents non-negative, not positive - zero is allowed.
Now try it yourself
Domain of h(x) = sqrt(5 - x)?
Check your answer
x <= 5.
Need 5 - x >= 0.
Test whether a relation is a function
Choose the method: Every input must have exactly one output.
EZT practice question
Is the relation {(1, 2), (1, 3)} a function?
Show the worked solution
- A function gives each input exactly one output.
- Here the input 1 is paired with both 2 and 3.
Answer: No - one input has two outputs, so it is not a function.
Check: Two inputs sharing one output is fine; one input splitting to two outputs is not.
Watch for this: Do not check whether outputs repeat - check whether inputs repeat with different partners.
Now try it yourself
Is {(1, 2), (2, 2)} a function?
Check your answer
Yes.
Each input has exactly one output; outputs may repeat.
Combine functions pointwise
Choose the method: (f + g)(x) = f(x) + g(x); (fg)(x) = f(x) g(x).
EZT practice question
Let f(x) = x^2 and g(x) = x + 1. Find (f + g)(2) and (fg)(2).
Show the worked solution
- f(2) = 4 and g(2) = 3.
- (f + g)(2) = 4 + 3 and (fg)(2) = 4 x 3.
Answer: (f + g)(2) = 7 and (fg)(2) = 12.
Check: Direct: (f + g)(x) = x^2 + x + 1 gives 4 + 2 + 1 = 7 - matches.
Watch for this: For f/g, exclude inputs where g(x) = 0 from the domain.
Now try it yourself
Same f and g - find (f - g)(2).
Check your answer
1.
f(2) - g(2) = 4 - 3 = 1.
Try a mixed set
Choose a method or explanation before opening each answer.
1. n(A x B)?
Show answer and reason
n(A) x n(B).
Each of n(A) first entries pairs with each of n(B) second entries.
2. Number of relations from A to B?
Show answer and reason
2^(n(A) n(B)).
Every subset of A x B is a relation.
3. Number of functions from A to B?
Show answer and reason
n(B)^n(A).
Each input independently chooses an output.
4. When is a relation a function?
Show answer and reason
When each input has exactly one output.
Outputs may repeat; inputs may not split.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- A = {0, 1}, B = {1, 2}: list all pairs in A x B.
- Find the domain of h(x) = sqrt(5 - x).
- Is {(1, 2), (1, 3)} a function? Give the reason.
Know the rule but lose track of the steps?
Bring your book edition, your school topic list and a question you attempted. Ask a tutor to explain the choice of method and test it with a different problem. Agree the lesson topics, schedule and fee before starting.
For cost context, see home tuition fees in Delhi NCR. Agree the topics, schedule and fee with your chosen tutor.
Questions about this practice page
- What is the difference between a relation and a function?
- A relation from A to B is any subset of A x B. A function is a relation with the extra rule that every input has exactly one output.
- What are domain, codomain and range?
- The domain is the set of inputs used, the codomain is the declared target set, and the range is the set of outputs actually hit - the range always sits inside the codomain.
- How do you count functions from A to B?
- Each of the n(A) inputs independently chooses one of n(B) outputs, giving n(B)^n(A) functions.
- Why does f/g sometimes have a smaller domain?
- Because the quotient is undefined wherever g(x) = 0, so those inputs must be excluded even if both f and g are defined there.
- Are these the NCERT exercise questions for Relations and Functions (Class 11)?
- No. EZT pages are original practice following the chapter's section order - use them to prepare for the textbook exercises, not as a substitute for your book.