Real-valued functions and algebra of functions - Question Bank

1. If f(x) = x and g(x) = x, what is (f o g)(x)?
A) x
B) x^2
C) 2x
D) 1
2. If f(x) = x and g(x) = c (constant), what is (f+g)(x)?
A) cx
B) x+c
C) xc
D) c
3. If f(x) = x^2 and g(x) = x^2, what is (f-g)(x)?
A) 0
B) x^4
C) 2x^2
D) 1
4. What is the range of f(x) = -x^2?
A) [0, inf)
B) (-inf, 0]
C) (-inf, inf)
D) (0, inf)
5. What is the domain of f(x) = sqrt(x^2 - 9)?
A) (-inf, -3] U [3, inf)
B) (-3, 3)
C) (-inf, inf)
D) (-3, inf)
6. If f(x) = x-1 and g(x) = x+1, what is the domain of (f/g)(x)?
A) All real numbers
B) All real numbers except 1
C) All real numbers except -1
D) (-inf, -1)
7. If f(x) = x+1 and g(x) = x-1, what is the domain of (f/g)(x)?
A) All real numbers
B) All real numbers except 1
C) All real numbers except -1
D) (-inf, 1)
8. If f(x) = 1/x and g(x) = x^2, what is the domain of (g o f)(x)?
A) All real numbers
B) All real numbers except 0
C) All positive real numbers
D) All non-negative real numbers
9. If f(x) = x^2 and g(x) = 1/x, what is the domain of (g o f)(x)?
A) All real numbers
B) All real numbers except 0
C) All positive real numbers
D) All non-negative real numbers
10. Let f(x) = sqrt(4-x^2). What is the range of f?
A) (-inf, 2]
B) [2, inf)
C) [0, 2]
D) (0, 2)
11. Let f(x) = sqrt(4-x^2). What is the domain of f?
A) (-inf, 2]
B) [2, inf)
C) [-2, 2]
D) (0, 2)
12. If f(x) = 0, what is its range?
A) All real numbers
B) The set {0}
C) The empty set
D) Positive real numbers
13. If f(x) = x and g(x) = x, then (f/g)(x) is defined for:
A) All real numbers
B) All real numbers except 0
C) All positive real numbers
D) All non-zero real numbers
14. What is the range of f(x) = log_a(x) where a > 0 and a != 1?
A) All positive real numbers
B) All non-negative real numbers
C) All real numbers
D) (-inf, 0)
15. What is the domain of f(x) = log_a(x) where a > 0 and a != 1?
A) All real numbers
B) All positive real numbers
C) All non-negative real numbers
D) All real numbers except 0
16. If f(x) = x^2 and g(x) = sqrt(x-a), what is the domain of (g o f)(x)?
A) All real numbers
B) All real numbers such that x^2 >= a
C) All non-negative real numbers
D) All real numbers except sqrt(a)
17. If f(x) = sqrt(x-a) and g(x) = x^2, what is the domain of (f o g)(x)?
A) All real numbers
B) All real numbers such that x^2 >= a
C) All non-negative real numbers
D) All real numbers except sqrt(a)
18. Let f(x) = x^2 + 1. Which of the following is NOT in the range of f?
A) 1
B) 2
C) 0
D) 5
19. If f(x) = 3x - 2, what is f^-1(x)?
A) (x+2)/3
B) (x-2)/3
C) 1/(3x-2)
D) 3x+2
20. What is the range of f(x) = 2^x?
A) All real numbers
B) All positive real numbers
C) All non-negative real numbers
D) (0, inf)
21. What is the domain of f(x) = 1/sqrt(x-2)?
A) (2, inf)
B) [2, inf)
C) (-inf, 2)
D) (-inf, 2]
22. If f(x) = x+1 and g(x) = x-1, what is (g o f)(x)?
A) x
B) x+2
C) x-2
D) x^2-1
23. If f(x) = x-1 and g(x) = x+1, what is (f o g)(x)?
A) x
B) x+2
C) x-2
D) x^2-1
24. If f(x) = x^2 - 1 and g(x) = x+1, what is the domain of (f/g)(x)?
A) All real numbers
B) All real numbers except -1
C) All real numbers except 1
D) (-inf, -1)
25. A function f is defined by f(x) = sqrt(1-x^2). What is its range?
A) (-inf, 1]
B) [1, inf)
C) [0, 1]
D) (0, 1)
26. A function f is defined by f(x) = sqrt(1-x^2). What is its domain?
A) (-inf, 1]
B) [1, inf)
C) [-1, 1]
D) (0, 1)
27. If f(x) = x^3, what is f^-1(x)?
A) x^(1/3)
B) 1/x^3
C) x
D) 3x
28. What is the range of f(x) = arccos(x)?
A) [0, pi]
B) [0, 1]
C) [-1, 1]
D) (0, pi)
29. What is the domain of f(x) = arcsin(x)?
A) (-inf, inf)
B) [0, 1]
C) [-1, 1]
D) (0, 1)
30. If f(x) = x^2 and g(x) = x-1, what is the domain of (f/g)(x)?
A) All real numbers
B) All real numbers except 1
C) All non-negative real numbers
D) [0, 1)
31. If f(x) = x+1 and g(x) = 1/(x+1), what is the domain of (f*g)(x)?
A) All real numbers
B) All real numbers except -1
C) All positive real numbers
D) (-1, inf)
32. If f(x) = 2 and g(x) = 3, what is (f+g)(x)?
A) 5
B) 6
C) f(x)+g(x)
D) 2+3
33. If f(x) = x and g(x) = x, then (f-g)(x) is:
A) x
B) 0
C) 2x
D) 1
34. Which of the following is NOT a real-valued function?
A) f(x) = x^2
B) f(x) = sqrt(x)
C) f(x) = 1/x
D) f(x) = sqrt(-x)
35. If f(x) = c (a constant function), what is its range?
A) All real numbers
B) All positive real numbers
C) The set {c}
D) The empty set
36. If f(x) = c (a constant function), what is its domain?
A) All positive real numbers
B) All non-negative real numbers
C) All real numbers
D) The set {c}
37. What is the range of f(x) = sin(x)?
A) [0, 1]
B) (-1, 1)
C) [-1, 1]
D) (-inf, inf)
38. What is the domain of f(x) = tan(x)?
A) All real numbers
B) All real numbers except n*pi, where n is an integer
C) All real numbers except (n + 1/2)*pi, where n is an integer
D) All real numbers except n*pi/2, where n is an integer
39. If f(x) = x^2 and g(x) = sqrt(x), what is the domain of (g o f)(x)?
A) All real numbers
B) All non-negative real numbers
C) All positive real numbers
D) (-inf, 0]
40. If f(x) = sqrt(x) and g(x) = x^2, what is the domain of (f o g)(x)?
A) All real numbers
B) All non-negative real numbers
C) All positive real numbers
D) (-inf, 0]
41. What is the range of f(x) = e^x?
A) All real numbers
B) All positive real numbers
C) All non-negative real numbers
D) (0, inf)
42. What is the domain of f(x) = ln(x)?
A) All real numbers
B) All positive real numbers
C) All non-negative real numbers
D) All real numbers except 0
43. If f(x) = x+1 and g(x) = x^2, what is (f o g)(x)?
A) x^2+1
B) (x+1)^2
C) x(x+1)
D) x^2(x+1)
44. If f(x) = x^2 and g(x) = x+1, what is (g o f)(x)?
A) (x+1)^2
B) x^2+1
C) x(x+1)
D) (x^2)+1
45. What is the range of f(x) = |x|?
A) All real numbers
B) All positive real numbers
C) All non-negative real numbers
D) All integers
46. What is the domain of f(x) = 1/x?
A) All real numbers
B) All real numbers except 0
C) All positive real numbers
D) All negative real numbers
47. If f(x) = 2x and g(x) = x + 3, what is (f/g)(x)?
A) (2x)/(x+3)
B) (x+3)/(2x)
C) 2x/(x+3)
D) 2x - (x+3)
48. If f(x) = x - 1 and g(x) = x^2, what is (f*g)(x)?
A) (x-1)x^2
B) x^2 - x
C) x^3 - x^2
D) x^2(x-1)
49. If f(x) = x + 1 and g(x) = 2x, what is (f+g)(x)?
A) 3x + 1
B) 2x^2 + x
C) x + 2x + 1
D) 2x + 1
50. What is the range of the function f(x) = x^2 + 1?
A) [0, inf)
B) [1, inf)
C) (-inf, 1]
D) (1, inf)
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