Q1. Let \( f(x) = \sqrt{17x – x^2 – 16} \). Find the domain of \( f(x) \).

• A) [-1, 1]
• B) [1, 4]
• C) (0, 16]
• D) [1, 16] ✅

Q2. The domain of \( f(x) = \sqrt{x – x^2} \) is:

• A) \( \mathbb{R} \)
• B) \( \mathbb{R} \setminus \{0\} \)
• C) (0, 1)
• D) [-1, 1] ✅

Q3. Let \( f(x) = |f(x)| + f(x) \), where \( f(x) = x^2 – 2 \) for \( x \in [0, 2] \), \( f(x) = -2 \) for \( x \in [-2, 0] \). Then \( f \) and \( g \) are:

• A) Injective
• B) Surjective ✅
• C) Bijective
• D) Not bijective

Q4. Let \( f(x) = 1 + 2x + 4x^2 + 8x^3 + \dots \), for \( |x| < \frac{1}{2} \). Find \( f^{-1}(x) \).

• A) \( \frac{x – 1}{2x} \) ✅
• B) \( \frac{x – 1}{x} \)
• C) \( \frac{1 – x}{2x} \)
• D) \( \frac{1 – 2x}{x} \)

Q5. Let \( f(x) = \tan\left(\frac{\pi x}{2}\right) \), and \( g(x) = \sqrt{3 + 4x – 4x^2} \). Find the domain of \( f + g \).

• A) (-1, 1)
• B) [1, 4]
• C) [-1, 1] ✅
• D) [1, 1]

Q6. If \( f(x) \) satisfies \( 2f(x) + f\left(\frac{1}{x}\right) = 4x \), and \( S = \{x \in \mathbb{R} \setminus \{0\} : f(x) = f(-x)\} \), then number of elements in \( S \) is:

• A) 0
• B) 1
• C) 2 ✅
• D) At least 3

Q7. Let \( f(x) = \frac{x}{x^2 + 1} \). Then \( f \) is:

• A) Injective
• B) Surjective
• C) Neither injective nor surjective ✅
• D) Bijective

Q8. If \( f(x) = \log_2(x + 1) \), then \( f^{-1}(x) = \)

• A) \( 2^x – 1 \)
• B) \( 2^x – 1 ✅ \)
• C) \( \log_2(x – 1) \)
• D) \( \log_2(x + 1) \)

Q9. Find the domain of \( f(x) = \frac{1}{\sqrt{x^2 – 4}} \)

• A) \( (-\infty, -2) \cup (2, \infty) \)
• B) \( (-\infty, -2) \cup (2, \infty) ✅ \)
• C) \( [-2, 2] \)
• D) \( \mathbb{R} \)

Q10. If \( f(x) = x^2 \), \( g(x) = \sqrt{x} \), then \( (g \circ f)(x) = \)

• A) \( \sqrt{x^2} \)
• B) \( x \)
• C) \( |x| ✅ \)
• D) \( x^2 \)

Q11. If \( f(x) = \frac{2x + 3}{x – 1} \), then \( f^{-1}(x) = \)

• A) \( \frac{x + 3}{2 – x} \)
• B) \( \frac{x – 3}{2 + x} \)
• C) \( \frac{x + 1}{2 – x} ✅ \)
• D) \( \frac{x – 1}{2 + x} \)

Q12. Find the domain of \( f(x) = \log_4(x^2 – 5x + 6) \)

• A) \( x \in \mathbb{R} \)
• B) \( x \in (2, 3) \)
• C) \( x \in (-\infty, 2) \cup (3, \infty) ✅ \)
• D) \( x \in [2, 3] \)

Q13. If \( f(x) = x + 1 \), \( g(x) = x^2 \), then \( (f \circ g)(x) = \)

• A) \( x^2 + 1 ✅ \)
• B) \( x + 1 \)
• C) \( x^2 + x \)
• D) \( x^2 + 2x + 1 \)

Q14. Let \( f(x) = \frac{1}{x} \), then \( f \) is:

• A) Injective ✅
• B) Surjective
• C) Bijective
• D) Not injective

Q15. If \( f(x) = x^3 + 1 \), then \( f^{-1}(x) = \)

• A) \( \sqrt[3]{x + 1} \)
• B) \( \sqrt[3]{x – 1} ✅ \)
• C) \( x^3 – 1 \)
• D) \( x – 1 \)

Q16. If \( f(x) = x + x^2 + x^3 + \dots + x^{2018} \), then \( f(0) + 1 = \)

• A) 0
• B) 1 ✅
• C) 2018
• D) Cannot be determined

Q17. Find the domain of \( f(x) = \frac{1}{\log(x)} \)

• A) \( x > 0 \)
• B) \( x > 0, x \ne 1 ✅ \)
• C) \( x \ne 0 \)
• D) \( x \in \mathbb{R} \)

Q18. If \( f(x) = x^2 \), \( g(x) = x + 1 \), then \( (f \circ g)(x) = \)

• A) \( x^2 + 1 \)
• B) \( x^2 + 2x + 1 ✅ \)
• C) \( x + 1 \)
• D) \( x^2 + x \)

Q19. Let \( f(x) = x^2 \), then \( f \) is:

• A) Injective
• B) Surjective
• C) Neither injective nor surjective ✅
• D) Bijective

Q20. If \( f(x) = \frac{1 – x}{1 + x} \), then \( f^{-1}(x) = \)

• A) \( \frac{1 – x}{1 + x} \)
• B) \( \frac{1 + x}{1 – x} \)
• C) \( \frac{1 – x}{1 + x} ✅ \)
• D) \( \frac{x – 1}{x + 1} \)

Q21. If \( f(x) = \frac{3x – 2}{x + 1} \), then \( f^{-1}(x) = \)

• A) \( \frac{2 + x}{3 – x} \)
• B) \( \frac{x + 2}{3 – x} \)
• C) \( \frac{2 + x}{3 – x} ✅ \)
• D) \( \frac{3x + 2}{1 – x} \)

Q22. Find the domain of \( f(x) = \frac{1}{\sqrt{4 – x^2}} \)

• A) \( (-2, 2) ✅ \)
• B) \( [-2, 2] \)
• C) \( \mathbb{R} \)
• D) \( \mathbb{R} \setminus \{0\} \)

Q23. Let \( f(x) = \sin x \), then \( f \) is:

• A) Injective
• B) Surjective
• C) Neither injective nor surjective ✅
• D) Bijective

Q24. If \( f(x) = \sqrt{x} \), \( g(x) = x^2 – 1 \), then \( (f \circ g)(x) = \)

• A) \( \sqrt{x^2 – 1} ✅ \)
• B) \( x^2 – 1 \)
• C) \( \sqrt{x^2 + 1} \)
• D) \( x – 1 \)

Q25. If \( f(x) = x^2 \), and \( f(x + 1) – f(x) = 7 \), then \( x = \)

• A) 2
• B) 3
• C) 3 ✅
• D) 4

Q26. The domain of \( f(x) = \log_4(x^2 – 4x + 3) \) is:

• A) \( (-\infty, 1) \cup (3, \infty) ✅ \)
• B) \( (1, 3) \)
• C) \( [1, 3] \)
• D) \( \mathbb{R} \)

Q27. If \( f(x) = \frac{1}{x – 1} \), then \( f^{-1}(x) = \)

• A) \( \frac{1}{x} + 1 \)
• B) \( \frac{1 + x}{x} \)
• C) \( \frac{1 + x}{x} ✅ \)
• D) \( \frac{x}{1 + x} \)

Q28. Let \( f(x) = \cos x \), then \( f \) is:

• A) Injective
• B) Surjective
• C) Neither injective nor surjective ✅
• D) Bijective

Q29. If \( f(x) = x^2 \), \( g(x) = \sqrt{x} \), then \( (f \circ g)(x) = \)

• A) \( x \)
• B) \( \sqrt{x^2} \)
• C) \( x ✅ \)
• D) \( |x| \)

Q30. Find the domain of \( f(x) = \frac{1}{\sqrt{1 – x^2}} \)

• A) \( (-1, 1) ✅ \)
• B) \( [-1, 1] \)
• C) \( \mathbb{R} \)
• D) \( \mathbb{R} \setminus \{0\} \)

Q31. If \( f(x) = \frac{1 + x}{1 – x} \), then \( f^{-1}(x) = \)

• A) \( \frac{x – 1}{x + 1} \)
• B) \( \frac{1 – x}{1 + x} \)
• C) \( \frac{x – 1}{x + 1} ✅ \)
• D) \( \frac{x + 1}{x – 1} \)

Q32. Find the domain of \( f(x) = \frac{1}{\sqrt{x^2 – 1}} \)

• A) \( (-\infty, -1) \cup (1, \infty) ✅ \)
• B) \( [-1, 1] \)
• C) \( \mathbb{R} \)
• D) \( \mathbb{R} \setminus \{0\} \)

Q33. If \( f(x) = x^2 \), \( g(x) = x + 1 \), then \( (g \circ f)(x) = \)

• A) \( x^2 + 1 ✅ \)
• B) \( x + 1 \)
• C) \( x^2 + x \)
• D) \( x^2 + 2x + 1 \)

Q34. Let \( f(x) = \frac{x^2}{x^2 + 1} \), then \( f \) is:

• A) Injective
• B) Surjective
• C) Neither injective nor surjective ✅
• D) Bijective

Q35. If \( f(x) = \frac{1}{x + 2} \), then \( f^{-1}(x) = \)

• A) \( \frac{1}{x} – 2 ✅ \)
• B) \( \frac{1}{x + 2} \)
• C) \( \frac{1}{x – 2} \)
• D) \( \frac{1}{x} + 2 \)

Q36. If \( f(x) = x^2 \), and \( f(x + 1) – f(x) = 9 \), then \( x = \)

• A) 2 ✅
• B) 3
• C) 4
• D) 5