Class 12 Maths · Original EZT practice
Relations and Functions (Class 12 Maths Chapter 1): the tests that decide one-one, onto and invertible
Everything in this chapter comes down to three questions. For a relation: is it reflexive, symmetric, transitive? For a function: is it one-one, is it onto? And only when both answers are yes does the function have an inverse. Below: the ideas in plain words, then original EZT practice with worked answers. (Note: this is original practice, not a textbook exercise key.)
10 original worked examples, 10 fresh self-check 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 17 October 2026:
1. Relations: the three tests
A relation R on a set A is a collection of ordered pairs from A. Reflexive: every element relates to itself. Symmetric: if a R b then b R a. Transitive: if a R b and b R c then a R c. A relation passing all three is an equivalence relation - it splits the set into clean groups.
2. Functions: one-one and onto
A function gives each input exactly one output. One-one (injective): different inputs always give different outputs. Onto (surjective): every value in the codomain is hit by some input. Both together = bijective.
3. Composition: functions feeding functions
f o g means "first g, then f": (f o g)(x) = f(g(x)). Order matters - f o g and g o f usually differ.
4. Invertible functions: only bijections
A function has an inverse exactly when it is one-one AND onto - then each output traces back to exactly one input. To find the inverse, write y = f(x), swap and solve for x.
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.
On A = {1, 2, 3}, R = {(1, 1), (2, 2), (3, 3)}. Check the three tests
Choose the method: A relation R on a set A is a collection of ordered pairs from A.
EZT practice question
On A = {1, 2, 3}, R = {(1, 1), (2, 2), (3, 3)}. Check the three tests.
Show the worked solution
- Name the set, relation or function rule being tested.
- Check the defining condition directly, and look for one counterexample before claiming it holds.
- State the conclusion with the domain or codomain attached.
Answer: reflexive (every element pairs with itself), symmetric (each pair reversed is itself), transitive (no chains to break it) - an equivalence relation.
Check: The substitution and final condition match the worked statement.
Watch for this: Do not stop after one positive example; a relation or function property must hold for every element in the stated domain.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
On natural numbers, "x divides y". Which tests pass?
Choose the method: A relation R on a set A is a collection of ordered pairs from A.
EZT practice question
On natural numbers, "x divides y". Which tests pass?
Show the worked solution
- Name the set, relation or function rule being tested.
- Check the defining condition directly, and look for one counterexample before claiming it holds.
- State the conclusion with the domain or codomain attached.
Answer: reflexive (x divides x), transitive (x|y and y|z give x|z), but NOT symmetric - 2 divides 4, yet 4 does not divide 2. Not an equivalence relation.
Check: The substitution and final condition match the worked statement.
Watch for this: Do not stop after one positive example; a relation or function property must hold for every element in the stated domain.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
Is "same remainder on division by 3" an equivalence relation on integers?
Choose the method: A relation R on a set A is a collection of ordered pairs from A.
EZT practice question
Is "same remainder on division by 3" an equivalence relation on integers?
Show the worked solution
- Name the set, relation or function rule being tested.
- Check the defining condition directly, and look for one counterexample before claiming it holds.
- State the conclusion with the domain or codomain attached.
Answer: yes - every number has its own remainder (reflexive), matching remainders is mutual (symmetric), and chains of matches carry over (transitive). It splits integers into three groups: remainders 0, 1 and 2.
Check: The substitution and final condition match the worked statement.
Watch for this: Do not stop after one positive example; a relation or function property must hold for every element in the stated domain.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
Is f(x) = 2x + 3 on the real numbers one-one?
Choose the method: A function gives each input exactly one output.
EZT practice question
Is f(x) = 2x + 3 on the real numbers one-one?
Show the worked solution
- Name the set, relation or function rule being tested.
- Check the defining condition directly, and look for one counterexample before claiming it holds.
- State the conclusion with the domain or codomain attached.
Answer: yes - if 2x1 + 3 = 2x2 + 3, subtracting and halving gives x1 = x2.
Check: The substitution and final condition match the worked statement.
Watch for this: Do not stop after one positive example; a relation or function property must hold for every element in the stated domain.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
Is f(x) = x^2 on the real numbers one-one or onto?
Choose the method: A function gives each input exactly one output.
EZT practice question
Is f(x) = x^2 on the real numbers one-one or onto?
Show the worked solution
- Name the set, relation or function rule being tested.
- Check the defining condition directly, and look for one counterexample before claiming it holds.
- State the conclusion with the domain or codomain attached.
Answer: neither - f(2) = f(-2) = 4 breaks one-one, and no input reaches y = -1, breaking onto.
Check: The substitution and final condition match the worked statement.
Watch for this: Do not stop after one positive example; a relation or function property must hold for every element in the stated domain.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
How many different functions go from {1, 2} to {a, b}?
Choose the method: A function gives each input exactly one output.
EZT practice question
How many different functions go from {1, 2} to {a, b}?
Show the worked solution
- Name the set, relation or function rule being tested.
- Check the defining condition directly, and look for one counterexample before claiming it holds.
- State the conclusion with the domain or codomain attached.
Answer: each of the 2 inputs has 2 choices, so 2 x 2 = 4 functions.
Check: The substitution and final condition match the worked statement.
Watch for this: Do not stop after one positive example; a relation or function property must hold for every element in the stated domain.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
f(x) = 2x, g(x) = x + 5. Find f o g and g o f
Choose the method: f o g means "first g, then f": (f o g)(x) = f(g(x)).
EZT practice question
f(x) = 2x, g(x) = x + 5. Find f o g and g o f.
Show the worked solution
- Name the set, relation or function rule being tested.
- Check the defining condition directly, and look for one counterexample before claiming it holds.
- State the conclusion with the domain or codomain attached.
Answer: f o g (x) = f(x + 5) = 2x + 10; g o f (x) = g(2x) = 2x + 5 - not the same.
Check: The substitution and final condition match the worked statement.
Watch for this: Do not stop after one positive example; a relation or function property must hold for every element in the stated domain.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
With the same f and g, find (f o g)(3)
Choose the method: f o g means "first g, then f": (f o g)(x) = f(g(x)).
EZT practice question
With the same f and g, find (f o g)(3).
Show the worked solution
- Name the set, relation or function rule being tested.
- Check the defining condition directly, and look for one counterexample before claiming it holds.
- State the conclusion with the domain or codomain attached.
Answer: g(3) = 8, then f(8) = 16.
Check: The substitution and final condition match the worked statement.
Watch for this: Do not stop after one positive example; a relation or function property must hold for every element in the stated domain.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
Find the inverse of f(x) = 2x + 3 on the reals
Choose the method: A function has an inverse exactly when it is one-one AND onto - then each output traces back to exactly one input.
EZT practice question
Find the inverse of f(x) = 2x + 3 on the reals.
Show the worked solution
- Name the set, relation or function rule being tested.
- Check the defining condition directly, and look for one counterexample before claiming it holds.
- State the conclusion with the domain or codomain attached.
Answer: y = 2x + 3 gives x = (y - 3)/2, so f-inverse(y) = (y - 3)/2. Check: f(f-inverse(7)) = f(2) = 7.
Check: The substitution and final condition match the worked statement.
Watch for this: Do not stop after one positive example; a relation or function property must hold for every element in the stated domain.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
Why does f(x) = x^2 on the reals have no inverse, while x^2 on x >= 0 do?
Choose the method: A function has an inverse exactly when it is one-one AND onto - then each output traces back to exactly one input.
EZT practice question
Why does f(x) = x^2 on the reals have no inverse, while x^2 on x >= 0 does?
Show the worked solution
- Name the set, relation or function rule being tested.
- Check the defining condition directly, and look for one counterexample before claiming it holds.
- State the conclusion with the domain or codomain attached.
Answer: on all reals it fails one-one (2 and -2 collide) and onto (negatives unreached); on x >= 0 every output has exactly one input, so the inverse exists: f-inverse(y) = sqrt(y). TRY IT (no answers, for practice): (a) Is "is a classmate of" reflexive, symmetric, transitive? (b) Show f(x) = 5x - 2 is bijective on the reals and find its inverse. (c) With f(x) = x + 1 and g(x) = 3x, find f o g (2). (d) How many functions go from a 3-member set to a 2-member set? Edition note: chapter numbering (1) matches the pre-rationalisation book; the current edition drops the binary-operations section, which this page does not cover. FAQ (FAQPage schema, 5 Q&A): Q1: What is an equivalence relation? A: a relation that is reflexive (everything relates to itself), symmetric (reversible) and transitive (chains carry over) - it partitions the set into disjoint groups. Q2: What is the difference between one-one and onto? A: one-one means no two inputs share an output; onto means every codomain value is an output of some input. A function can be either, both, or neither. Q3: When is a function invertible? A: exactly when it is bijective - one-one and onto - so each output maps back to one input; find the inverse by solving y = f(x) for x. Q4: Why does composition order matter? A: f o g applies g first, then f - changing the order usually changes the result, as with 2(x + 5) versus 2x + 5. Q5: Are these the NCERT exercise questions? A: No. EZT pages are original practice on this chapter - use them to prepare for the textbook exercises, not as a substitute for your book. Internal links: /class-11-maths-chapter-1-sets (live), /class-10-maths-chapter-2-polynomials (live), /tuitions, /home-tuition-fees-delhi-ncr Schema: LearningResource + FAQPage, same component pattern as existing solutions pages Mobile: responsive
Check: The substitution and final condition match the worked statement.
Watch for this: Do not stop after one positive example; a relation or function property must hold for every element in the stated domain.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- Is "is a classmate of" reflexive, symmetric, transitive?
- Show f(x) = 5x - 2 is bijective on the reals and find its inverse.
- With f(x) = x + 1 and g(x) = 3x, find f o g (2).
- How many functions go from a 3-member set to a 2-member set?
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Questions about this practice page
- What is an equivalence relation?
- a relation that is reflexive (everything relates to itself), symmetric (reversible) and transitive (chains carry over) - it partitions the set into disjoint groups.
- What is the difference between one-one and onto?
- one-one means no two inputs share an output; onto means every codomain value is an output of some input. A function can be either, both, or neither.
- When is a function invertible?
- exactly when it is bijective - one-one and onto - so each output maps back to one input; find the inverse by solving y = f(x) for x.
- Why does composition order matter?
- f o g applies g first, then f - changing the order usually changes the result, as with 2(x + 5) versus 2x + 5.
- Are these the NCERT exercise questions for Relations and Functions?
- No. EZT pages are original practice on this chapter - use them to prepare for the textbook exercises, not as a substitute for your book.