Functions: EAPCET PYQs (2020–2023) – Topic Wise
Expert’s Note: Mastering Functions is about understanding definitions and practicing their application. This compilation is designed to strengthen your core concepts and problem-solving speed. Focus on understanding the “why” behind each solution.
Topic 1: Domain of a Function
The set of all possible input values (x) for which the function is defined.
Key Concepts to Remember:
- √(f(x)) requires f(x) ≥ 0.
- 1/f(x) requires f(x) ≠ 0.
- log(f(x)) requires f(x) > 0.
- For sin⁻¹(f(x)) and cos⁻¹(f(x)), we need -1 ≤ f(x) ≤ 1.
[2020 Shift-1] Find the domain of f(x) = cos⁻¹((x – 3)/2) − log₁₀(4 − x).
Solution: For cos⁻¹: −1 ≤ (x−3)/2 ≤ 1 → 1 ≤ x ≤ 5. For log: 4−x>0 → x<4.
Domain: [1, 4)
[2020 Shift-2] Domain of √(|x|−x).
Solution: |x|−x ≥0 → true for x ≤ 0.
Domain: (−∞, 0]
[2021 Shift-1] Domain of f(x) = sec⁻¹(3x − 4) + tanh⁻¹((x + 3)/5).
Solution: |3x−4| ≥1 and −1 < (x+3)/5 < 1.
Domain: (−8, 1] ∪ [5/3, 2)
[2021 Shift-1] Domain of √(log₁₀((5x − x²)/4)).
Solution: (5x−x²)/4 > 0 and log ≥ 0 → x ∈ [1,4].
Domain: [1, 4]
[2022 Shift-1] Domain of f(x) = log₂(x + 3) / √(x² + 3x + 2).
Solution: x > −3 and x² + 3x + 2 > 0 → (x+1)(x+2) > 0.
Domain: (−3, −2) ∪ (−1, ∞)
[2022 Shift-2] Domain of (√(2−x) + √(1+x)) / √(x+3).
Domain: [−1, 2]
[2023 Shift-1] Domain of log₀.₅(x−3) / √(x−1).
Domain: (3, 4]
Topic 2: Range of a Function
- For quadratic functions, complete the square.
- For rational functions, express x in terms of y.
- For √ expressions, check the limits of the inside term.
[2020 Shift-2] Range of f(x) = x⁶ / (x⁶ + 2020).
Range: [0, 1)
[2021 Shift-1] Range of f(x) = x² + 1/(x² + 1).
Range: [1, ∞)
[2022 Shift-1] Range of f(x) = (x² + x + 1)/x.
Range: (−∞, −1] ∪ [3, ∞)
[2023 Shift-1] Range of |x−2| + |x−3| → [1, ∞)
[2023 Shift-1] Range of √(9−x²) → [0, 3]
Topic 3: Types of Functions (Injective, Surjective, Bijective)
[2020 Shift-2] f(x) = x / √(1+x²) → Injective but not surjective.
[2021 Shift-1] For finite S, non-identity function can be bijective.
[2022 Shift-2] f((m,n)) = 2^(m−1)(2n−1) → Bijective.
[2023 Shift-1] f(x) = x−[x]+3 → Periodic with period 1.
Topic 4: Functional Equations
[2020 Shift-1] Bijective functions on Z satisfying f(x+y)=f(x)+f(y):
f(x)=x or f(x)=−x
[2020 Shift-2] f(x+y)=f(x)+f(y) → f(x)=kx (Infinitely many such functions)
[2020 Shift-1] f(1)=7, f(x+y)=f(x)+f(y) → f(x)=7x → Σf(t)=3255
[2020 Shift-2] (f(x))²=f(x²)+f(1) → f(x)=x+1/x
[2021 Shift-2] f(x)=x is one-one but not onto.
Topic 5: Inverse & Composition of Functions
[2020 Shift-1] f(t)=3t−2, (g∘f)⁻¹(t)=t−2 → g(t)=(t+8)/3
[2021 Shift-2] Inverse of y=(10ˣ−10⁻ˣ)/(10ˣ+10⁻ˣ) → ½log₁₀((1+x)/(1−x))
[2022 Shift-1] f(x)=(x+2)²−2, x≥−2 → f⁻¹(x)=√(x+2)−2
Topic 6: Greatest Integer Function [x]
[2020 Shift-1] f(x)=[x]²−[x]−2>0 → Domain: R−[−1,3)
[2020 Shift-2] f(x)=sinπ[x]/(1+[x])+x/(2+3x) → Domain: R−[−1,−2/3), Range: R−{1/3}
[2021 Shift-2] f(x)=[x], g(x)=3[x/3] → x∈[3k,3k+1), k∈Z
Quick Answer Key
1. [1, 4) 2. (−∞, 0] 3. (−8, 1] ∪ [5/3, 2) 4. [1, 4] 5. (−3, −2) ∪ (−1, ∞) 6. [−1, 2] 7. (3, 4]
8. [0, 1) 9. [1, ∞) 10. (−∞, −1] ∪ [3, ∞) 11. [1, ∞) 12. [0, 3] 13. Injective not surjective 14. Bijective 15. Bijective 16. Periodic (1) 17. 2 18. ∞ 19. 3255 20. x+1/x 21. One-one not onto 22. g(t)=(t+8)/3 23. ½log₁₀((1+x)/(1−x)) 24. √(x+2)−2 25. R−[−1,3) 26. R−[−1,−2/3), R−{1/3} 27. x∈[3k,3k+1)
— AIMSTUTORIAL
