Relations and Functions — Class 12 MCQs with Answers

Relations and Functions — important questions & MCQs with answers (Class 12 Mathematics)

14 solved questions from this chapter's difficulty levels, each with its answer and explanation. The other 76 are timed and scored when you sign in.

1. Q1Easy

   Relation R = {(a,a) : a ∈ A} is:

   A.Symmetric only

   B.Reflexive✓ Correct

   C.Transitive only

   D.Empty

   Answer: B. Reflexive

   Explanation: R = {(a,a) : a ∈ A} relates every element only to itself, matching the definition of reflexive exactly; it makes no claim about symmetry or transitivity.

2. Q2Easy

   f(x) = x is:

   A.Not function

   B.Many-one

   C.Constant

   D.Bijective on R✓ Correct

   Answer: D. Bijective on R

   Explanation: f(x)=x is the identity function: distinct inputs give distinct outputs (one-one) and every real y is hit by x=y (onto), so it is bijective on R.

3. Q3Easy

   A relation R on set A is reflexive if:

   A.(a, b) ∈ R ⇒ (b, a) ∈ R

   B.(a, a) ∈ R for all a ∈ A✓ Correct

   C.Empty

   D.Universal

   Answer: B. (a, a) ∈ R for all a ∈ A

   Explanation: Reflexive means (a,a) ∈ R for every a ∈ A — every element must relate to itself, unlike symmetric which swaps existing pairs (a,b)→(b,a).

4. Q4Easy

   A function f is one-one if:

   A.f is constant

   B.f(x₁) = f(x₂) ⇒ x₁ = x₂✓ Correct

   C.Every element has same image

   D.Many-to-one

   Answer: B. f(x₁) = f(x₂) ⇒ x₁ = x₂

   Explanation: One-one (injective) means f(x₁)=f(x₂) forces x₁=x₂ — distinct inputs must give distinct outputs; it says nothing about every output being constant.

5. Q5Easy

   (g ∘ f)(x) means:

   A.g(x) + f(x)

   B.f(g(x))

   C.g(x) · f(x)

   D.g(f(x))✓ Correct

   Answer: D. g(f(x))

   Explanation: (g∘f)(x) means apply f first, then g to that result: g(f(x)) — the rightmost function in the notation acts first, not g(x)+f(x) or a product.

6. Q6Easy

   A function has an inverse iff it is:

   A.Onto only

   B.One-one only

   C.Bijective✓ Correct

   D.Constant

   Answer: C. Bijective

   Explanation: A function has an inverse exactly when it's bijective: one-one so outputs trace to a unique input, and onto so every codomain element has a pre-image at all.

7. Q7Medium

   Equivalence relation needs:

   A.Only reflexive

   B.Reflexive, symmetric, transitive✓ Correct

   C.Antisymmetric

   D.Only transitive

   Answer: B. Reflexive, symmetric, transitive

   Explanation: An equivalence relation must be reflexive, symmetric, and transitive all at once — having only one or two of these, like just reflexive, is not enough.

8. Q8Medium

   If f(x)=2x, g(x)=x+1, (f∘g)(3) =

   A.8✓ Correct

   B.7

   C.6

   D.4

   Answer: A. 8

   Explanation: Apply g first: g(3)=3+1=4, then apply f to that result: f(4)=2(4)=8 — the rightmost function g always acts on x before the leftmost function f.

9. Q9Medium

   R on N defined by aRb iff a = b is:

   A.Not equivalence

   B.Symmetric only

   C.Transitive only

   D.Equivalence✓ Correct

   Answer: D. Equivalence

   Explanation: Equality aRb: a=b is reflexive (a=a), symmetric (a=b⇒b=a), and transitive (a=b,b=c⇒a=c) — satisfying all three makes it an equivalence relation, not just symmetric alone.

10. Q10Medium

    If f(x) = √x and g(x) = x², (f ∘ g)(x) =

    A.x

    B.|x|✓ Correct

    C.x²

    D.1/x

    Answer: B. |x|

    Explanation: f(g(x)) = √(x²) = |x|, not x — squaring first destroys the sign, and the square root always returns the non-negative root, so negative x still maps to a positive value.

11. Q11Medium

    If f(x) = (x + 1)/(x − 1), x ≠ 1, then (f ∘ f)(x) =

    A.−x

    B.1

    C.x✓ Correct

    D.1/x

    Answer: C. x

    Explanation: Substituting f(x)=(x+1)/(x−1) into itself simplifies to [2x/(x−1)] ÷ [2/(x−1)] = x, so f is an involution — applying it twice returns the original input.

12. Q12Hard

    f has an inverse iff it is:

    A.Bijective✓ Correct

    B.Onto only

    C.One-one only

    D.Constant

    Answer: A. Bijective

    Explanation: A function has an inverse exactly when it's bijective: one-one so each output traces to a unique input, and onto so every codomain element has a pre-image to trace back to.

13. Q13Hard

    R on R defined by aRb iff a² + b² = 1. R is:

    A.Transitive

    B.Reflexive

    C.Equivalence

    D.Symmetric✓ Correct

    Answer: D. Symmetric

    Explanation: a²+b²=1 gives b²+a²=1 automatically, so R is symmetric; but a²+a²=1 fails for most a (not reflexive), and it doesn't chain to give transitivity either.

14. Q14Hard

    f: R → R defined by f(x) = x|x| is:

    A.Many-one

    B.Bijective✓ Correct

    C.Not onto

    D.Constant

    Answer: B. Bijective

    Explanation: f(x)=x|x| equals x² for x≥0 and −x² for x<0, and this combination is strictly increasing across all of R, so it never repeats a value and its range covers all of R — bijective.

Relations and Functions — FAQs

More Class 12 Mathematics chapters

• 02Inverse Trigonometric Functions
• 03Matrices
• 04Determinants
• 05Continuity and Differentiability
• 06Application of Derivatives
• 07Integrals

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