Classification of real valued functions MCQs With Answer

Classification of real valued functions MCQs With Answer is a focused study resource designed for B.Pharm students to master real-valued functions used in pharmaceutical calculations and modeling. This introduction covers classification of real-valued functions, including injective, surjective, bijective, even, odd, continuous, discontinuous, monotonic, polynomial, rational and trigonometric functions, emphasizing domain, range, and graphical interpretation. Targeted MCQs with answers reinforce definitions, properties, inverse and composite functions, limits, and common discontinuities, helping pharmacy students improve problem-solving speed and exam performance. Practice these concept-driven questions to build mathematical rigor relevant to pharmaceutics and pharmacokinetic modeling. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What defines a real-valued function?

• A mapping from a set to the set of real numbers
• A mapping from real numbers to complex numbers
• A mapping from integers to real numbers only
• A mapping that outputs only positive numbers

Correct Answer: A mapping from a set to the set of real numbers

Q2. For f(x)=x^2, what is the range when domain is all real numbers?

• All real numbers
• All non-negative real numbers
• All positive real numbers
• All non-positive real numbers

Correct Answer: All non-negative real numbers

Q3. Which property defines an injective (one-to-one) function?

• Different inputs may give same output
• Each output corresponds to exactly one input
• Function is periodic
• Function is continuous everywhere

Correct Answer: Each output corresponds to exactly one input

Q4. Which function is surjective onto real numbers?

• f(x)=e^x
• f(x)=x^3
• f(x)=sin x
• f(x)=x^2

Correct Answer: f(x)=x^3

Q5. When does a function have an inverse function?

• When it is continuous
• When it is bijective on its domain
• When it is differentiable
• When it is bounded

Correct Answer: When it is bijective on its domain

Q6. f(x)=x^3 is which of the following?

• Even function
• Odd function
• Neither even nor odd
• Periodic function

Correct Answer: Odd function

Q7. Which definition matches an even function?

• f(-x)=f(x) for all x in domain
• f(-x)=-f(x) for all x in domain
• f(x+T)=f(x) for some T
• f(x)≥0 for all x

Correct Answer: f(-x)=f(x) for all x in domain

Q8. A function is strictly increasing on an interval if:

• f(x1) ≤ f(x2) whenever x1 < x2
• f(x1) < f(x2) whenever x1 < x2
• f′(x)=0 everywhere
• f has no maxima

Correct Answer: f(x1) < f(x2) whenever x1 < x2

Q9. Which of these is a periodic function?

• f(x)=x^2
• f(x)=sin x
• f(x)=e^x
• f(x)=ln x

Correct Answer: f(x)=sin x

Q10. The domain of f(x)=ln(x-1) is:

• (-∞, ∞)
• (0, ∞)
• (1, ∞)
• [-1, ∞)

Correct Answer: (1, ∞)

Q11. For f(x)=1/x, what type of discontinuity exists at x=0?

• Removable discontinuity
• Jump discontinuity
• Infinite (essential) discontinuity
• Continuous

Correct Answer: Infinite (essential) discontinuity

Q12. A removable discontinuity occurs when:

• Left and right limits differ
• Limit exists but function value is different or undefined
• Function tends to infinity
• Function oscillates without limit

Correct Answer: Limit exists but function value is different or undefined

Q13. The floor function ⌊x⌋ is discontinuous at which points?

• All integers
• All real numbers
• No points; it is continuous
• Only at half-integers

Correct Answer: All integers

Q14. Which of the following is necessary (but not sufficient) for differentiability at a point?

• Function is bounded
• Function is continuous at that point
• Function is periodic
• Function has an inverse

Correct Answer: Function is continuous at that point

Q15. If f is even and g is odd, f(g(x)) is generally:

• Even
• Odd
• Neither
• Both even and odd

Correct Answer: Even

Q16. The composition (g∘f)(x) is defined when:

• Domain of f is subset of domain of g
• Range of f is subset of domain of g
• Domain of g equals domain of f
• f and g are both continuous

Correct Answer: Range of f is subset of domain of g

Q17. Which function is unbounded on the real line?

• f(x)=sin x
• f(x)=x
• f(x)=e^{-x^2}
• f(x)=cos x

Correct Answer: f(x)=x

Q18. The range of f(x)=tan x restricted to (-π/2, π/2) is:

• (-1,1)
• All real numbers
• [0, ∞)
• (-∞,0]

Correct Answer: All real numbers

Q19. Which of the following best describes a bijection?

• Injective but not surjective
• Surjective but not injective
• Both injective and surjective
• Neither injective nor surjective

Correct Answer: Both injective and surjective

Q20. If f(x)=x^2 and domain is restricted to x≥0, f becomes:

• Not defined
• Injective
• Surjective onto all reals
• Odd function

Correct Answer: Injective

Q21. A function f is bounded below if:

• There exists M such that f(x) ≤ M for all x
• There exists m such that f(x) ≥ m for all x
• f(x) takes only finite values
• f(x) has no maxima

Correct Answer: There exists m such that f(x) ≥ m for all x

Q22. Which statement about continuous functions on a closed interval [a,b] is True?

• They always have a maximum and minimum on [a,b]
• They may be unbounded on [a,b]
• They cannot be integrable
• They must be differentiable

Correct Answer: They always have a maximum and minimum on [a,b]

Q23. The derivative f′(a) gives information about:

• The integral of f
• The slope of the tangent at x=a
• The range of f
• The periodicity of f

Correct Answer: The slope of the tangent at x=a

Q24. Which function is continuous everywhere but not differentiable at x=0?

• f(x)=|x|
• f(x)=x^2
• f(x)=e^x
• f(x)=sin x

Correct Answer: f(x)=|x|

Q25. For f(x)=1/(x-2), what is the domain?

• All real numbers
• All real numbers except x=0
• All real numbers except x=2
• (2, ∞)

Correct Answer: All real numbers except x=2

Q26. Which of the following is a rational function?

• f(x)=sin x
• f(x)=x^2/(x+1)
• f(x)=e^x
• f(x)=ln x

Correct Answer: f(x)=x^2/(x+1)

Q27. Which concept helps determine horizontal asymptotes of a rational function?

• Behavior of f near vertical asymptotes
• Degree comparison of numerator and denominator
• Continuity at x=0
• Function parity

Correct Answer: Degree comparison of numerator and denominator

Q28. A function f has a jump discontinuity at x=a if:

• Left and right limits exist and are unequal
• Both left and right limits are infinite
• Limit exists and equals function value
• Function is differentiable at a

Correct Answer: Left and right limits exist and are unequal

Q29. Which of these functions is odd?

• f(x)=cos x
• f(x)=x^5
• f(x)=x^2+1
• f(x)=|x|

Correct Answer: f(x)=x^5

Q30. The domain of f(x)=√(x+3) is:

• x ≥ -3
• x > -3
• All real numbers
• x ≤ -3

Correct Answer: x ≥ -3

Q31. Which property ensures a continuous inverse on an interval?

• Function is strictly monotonic and continuous
• Function is bounded
• Function has even symmetry
• Function is periodic

Correct Answer: Function is strictly monotonic and continuous

Q32. If lim[x→a+] f(x)=L1 and lim[x→a-] f(x)=L2 with L1≠L2, then f is:

• Continuous at a
• Has a removable discontinuity at a
• Has a jump discontinuity at a
• Differentiable at a

Correct Answer: Has a jump discontinuity at a

Q33. The function f(x)=e^x is:

• Periodic
• Bounded
• Continuous and differentiable for all real x
• Only defined for x>0

Correct Answer: Continuous and differentiable for all real x

Q34. Which of the following statements about even and odd parts of a function is correct?

• Every function can be written as sum of an even and an odd function
• Only polynomials have even and odd parts
• Even and odd parts are always zero
• Decomposition is unique only for continuous functions

Correct Answer: Every function can be written as sum of an even and an odd function

Q35. The smallest positive period of sin(2x) is:

• π
• 2π
• π/2
• 4π

Correct Answer: π

Q36. If f′(x)>0 on an interval, then f is:

• Strictly increasing on that interval
• Strictly decreasing on that interval
• Constant on that interval
• Not defined on that interval

Correct Answer: Strictly increasing on that interval

Q37. Which function has an essential oscillatory discontinuity at x=0?

• f(x)=sin(1/x) for x≠0, f(0)=0
• f(x)=1/x
• f(x)=x^2
• f(x)=|x|

Correct Answer: f(x)=sin(1/x) for x≠0, f(0)=0

Q38. For which of the following is the horizontal asymptote y=0?

• f(x)=x^2/(x+1)
• f(x)=1/x
• f(x)=x
• f(x)=sin x

Correct Answer: f(x)=1/x

Q39. The function f(x)=arctan x has which range?

• (-π/2, π/2)
• [0, π]
• (-∞, ∞)
• [-π/2, π/2]

Correct Answer: (-π/2, π/2)

Q40. If f is continuous and f(a)·f(b) < 0, then by Intermediate Value Theorem:

• f has a discontinuity between a and b
• There exists c in (a,b) with f(c)=0
• f is differentiable on (a,b)
• f is monotonic on [a,b]

Correct Answer: There exists c in (a,b) with f(c)=0

Q41. Which of these is true for a strictly monotone function on an interval?

• It must be periodic
• It is one-to-one on that interval
• It has a finite number of discontinuities
• Its derivative is zero everywhere

Correct Answer: It is one-to-one on that interval

Q42. For f(x)=x/(1+x^2), maximum absolute value occurs approximately at:

• x=0
• x=1
• x=1/√3
• x→∞

Correct Answer: x=1/√3

Q43. Which of the following functions is not elementary algebraic?

• f(x)=√x
• f(x)=x^3
• f(x)=e^x
• f(x)=1/(x+1)

Correct Answer: f(x)=e^x

Q44. The function f(x)=0 for x rational and 1 for x irrational is:

• Continuous everywhere
• Discontinuous everywhere
• Continuous at rational points only
• Differentiable everywhere

Correct Answer: Discontinuous everywhere

Q45. The range of f(x)=x/(x^2+1) for real x is:

• All real numbers
• (-1/2, 1/2)
• (-∞, ∞)
• [0,1]

Correct Answer: (-1/2, 1/2)

Q46. If f(x)=g(x) for all x except at a single point where values differ, then:

• They have different limits everywhere
• They are equal as functions
• They have the same limit at every point where limit exists
• They must have different domains

Correct Answer: They have the same limit at every point where limit exists

Q47. Which operation can change injectivity of a function?

• Restricting the domain
• Composing with an injective function on appropriate range
• Both restricting domain and composing carefully
• None; injectivity is invariant

Correct Answer: Both restricting domain and composing carefully

Q48. A continuous function mapping compact set to real numbers is always:

• Unbounded
• Attains its bounds and is bounded
• Not integrable
• Discontinuous outside

Correct Answer: Attains its bounds and is bounded

Q49. For a function f, if lim[x→∞] f(x)=L finite, then L is called:

• Vertical asymptote
• Horizontal asymptote
• Removable discontinuity
• Essential singularity

Correct Answer: Horizontal asymptote

Q50. The domain of the composite function g∘f is:

• All real numbers always
• Set of x in domain of f such that f(x) is in domain of g
• Intersection of domains of f and g
• Union of ranges of f and g

Correct Answer: Set of x in domain of f such that f(x) is in domain of g

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