Real Valued function MCQs With Answer

Real Valued function MCQs With Answer are essential for B. Pharm students to master mathematical foundations used in pharmacokinetics, dose–response modeling, and analytical method calibration. This concise, keyword-rich introduction covers real-valued functions, domain and range, continuity, differentiability, limits, monotonicity, and practical applications in drug concentration–time profiles. These targeted MCQs help B. Pharm students strengthen problem-solving skills for exam preparation and real-world pharmaceutical modeling. Each question links core mathematical concepts to pharmaceutical contexts like half-life calculations and standard curves, enhancing conceptual clarity and application. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is a real-valued function?

• A mapping that assigns each element of a set to a real number
• A mapping that assigns each real number to a complex number
• A mapping that assigns each complex number to a matrix
• A mapping that assigns each integer to a polynomial

Correct Answer: A mapping that assigns each element of a set to a real number

Q2. In the context of real-valued functions, what is the domain?

• The set of all possible output values
• The set of all possible input values
• The set of critical points
• The set of asymptotes

Correct Answer: The set of all possible input values

Q3. What is the range (codomain) of a real-valued function?

• The set of all possible input values
• The set of all possible output values
• The set of derivative values only
• The set of integrals only

Correct Answer: The set of all possible output values

Q4. Which of the following is a real-valued function on all real numbers?

• f(x) = sqrt(x)
• f(x) = 1/x
• f(x) = x^3 + 2x
• f(x) = ln(x)

Correct Answer: f(x) = x^3 + 2x

Q5. When is a function called injective (one-to-one)?

• When distinct inputs can map to the same output
• When each output has at least two inputs
• When distinct inputs always map to distinct outputs
• When the range equals the domain

Correct Answer: When distinct inputs always map to distinct outputs

Q6. What does surjective (onto) mean for a real-valued function?

• Every element of the codomain is an image of at least one domain element
• Every element of the domain is an image of the codomain
• The function is one-to-one
• The function is periodic

Correct Answer: Every element of the codomain is an image of at least one domain element

Q7. A bijective function is:

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

Correct Answer: Both injective and surjective

Q8. If f and g are real-valued functions, which is true about composition f∘g?

• Domain of f∘g is always the same as domain of f
• f∘g is defined when range of g lies in domain of f
• f∘g is only defined for linear functions
• f∘g is always invertible

Correct Answer: f∘g is defined when range of g lies in domain of f

Q9. When does a real-valued function have an inverse function?

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

Correct Answer: When it is bijective on its domain

Q10. An even function satisfies which property?

• f(-x) = -f(x)
• f(-x) = f(x)
• f(x+T)=f(x)
• f'(x)=0 for all x

Correct Answer: f(-x) = f(x)

Q11. A periodic real-valued function has which characteristic?

• It is defined only on integers
• There exists T>0 such that f(x+T)=f(x) for all x
• Its derivative is always zero
• It has no maximum value

Correct Answer: There exists T>0 such that f(x+T)=f(x) for all x

Q12. Consider f(x)=x^2 for x≤1 and f(x)=2x-1 for x>1. Is f continuous at x=1?

• Yes, because left and right limits equal 1
• No, because left limit is 1 and right limit is 2
• No, because function is not defined at 1
• Yes, because derivative exists at 1

Correct Answer: Yes, because left and right limits equal 1

Q13. Which statement about limits is true for sums?

• Limit of a sum equals sum of limits if both limits exist
• Limit of a sum equals product of limits
• Limit of a sum does not depend on individual limits
• Sum of limits exists only for continuous functions

Correct Answer: Limit of a sum equals sum of limits if both limits exist

Q14. The derivative f'(a) is defined as:

• The integral from 0 to a of f(x)
• The limit as h→0 of [f(a+h)-f(a)]/h
• f(a+1) – f(a-1)
• The maximum of f on [a-1,a+1]

Correct Answer: The limit as h→0 of [f(a+h)-f(a)]/h

Q15. What is the derivative of e^x?

• e^x
• x e^{x-1}
• ln(x)
• 1/x

Correct Answer: e^x

Q16. What is the derivative of ln(x) for x>0?

• ln(x)^2
• 1/x
• e^x
• x ln(x)

Correct Answer: 1/x

Q17. A critical point of a real-valued function is where:

• The function is discontinuous
• The derivative is zero or undefined
• The function equals zero only
• The function has an asymptote

Correct Answer: The derivative is zero or undefined

Q18. A local maximum at x=c means:

• f(c) is greater than or equal to f(x) for x near c
• f(c) is the largest value on the entire domain
• f'(c) is always positive
• f(c) equals zero

Correct Answer: f(c) is greater than or equal to f(x) for x near c

Q19. If f'(x) > 0 for all x in an interval, then f is:

• Constant on that interval
• Decreasing on that interval
• Increasing on that interval
• Oscillatory on that interval

Correct Answer: Increasing on that interval

Q20. A bounded real-valued function means:

• Its derivative is bounded
• Its values lie between two finite numbers
• Its domain is finite
• It has no zeros

Correct Answer: Its values lie between two finite numbers

Q21. Which of the following is an unbounded function on ℝ?

• f(x) = sin(x)
• f(x) = e^{-x}
• f(x) = x^3
• f(x) = 1/(1+x^2)

Correct Answer: f(x) = x^3

Q22. Which theorem states a continuous function on a closed interval attains its maximum and minimum?

• Mean Value Theorem
• Intermediate Value Theorem
• Weierstrass (Extreme Value) Theorem
• Rolle’s Theorem

Correct Answer: Weierstrass (Extreme Value) Theorem

Q23. Uniform continuity differs from pointwise continuity in that:

• Uniform continuity requires the same δ for entire domain given ε
• Uniform continuity only applies at a single point
• Pointwise continuity is stronger than uniform continuity
• Uniform continuity requires differentiability

Correct Answer: Uniform continuity requires the same δ for entire domain given ε

Q24. Is every differentiable function continuous?

• No, differentiability and continuity are unrelated
• Yes, differentiability implies continuity
• Only if the function is polynomial
• Only if the function is bounded

Correct Answer: Yes, differentiability implies continuity

Q25. Does continuity imply differentiability?

• Yes, always
• No, continuity does not guarantee differentiability
• Only for linear functions
• Only if function is periodic

Correct Answer: No, continuity does not guarantee differentiability

Q26. Which of the following is true about polynomial functions?

• They are continuous and differentiable everywhere
• They are discontinuous at integer points
• They are only defined for positive x
• They always have horizontal asymptotes

Correct Answer: They are continuous and differentiable everywhere

Q27. If f and g are continuous, then f∘g is:

• Not defined
• Always discontinuous
• Continuous
• Always constant

Correct Answer: Continuous

Q28. What is lim_{x→∞} 1/x ?

• 1
• 0
• ∞
• Does not exist

Correct Answer: 0

Q29. What is lim_{x→∞} e^{-x} ?

• ∞
• 0
• 1
• −∞

Correct Answer: 0

Q30. Which function is not real-valued for negative x?

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

Correct Answer: f(x) = sqrt(x)

Q31. If f is even and differentiable, what is f'(0)?

• Undefined
• Equal to f(0)
• Equal to 0
• Equal to 1

Correct Answer: Equal to 0

Q32. The integral from −a to a of an odd function is:

• Twice the integral from 0 to a
• Zero
• Undefined
• Equal to the integral from 0 to a

Correct Answer: Zero

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

• (−∞, ∞)
• (0, ∞)
• [0, ∞)
• (−1, 1)

Correct Answer: (0, ∞)

Q34. The natural logarithm ln(x) has which range?

• [0, ∞)
• (−∞, ∞)
• (0, 1)
• (−1, 1)

Correct Answer: (−∞, ∞)

Q35. Is the exponential function e^x injective on ℝ?

• No, it is periodic
• Yes, because it is strictly increasing
• No, it maps many x to same y
• Only on positive reals

Correct Answer: Yes, because it is strictly increasing

Q36. What is the fundamental period of sin(x)?

• π
• 2π
• π/2
• 1

Correct Answer: 2π

Q37. If f has an inverse f^{-1}, then f∘f^{-1} equals:

• The zero function
• f
• The identity function on the codomain of f
• The derivative of f

Correct Answer: The identity function on the codomain of f

Q38. A function f is continuous at a if:

• lim_{x→a} f(x) exists and equals f(a)
• f'(a) exists
• f has a vertical asymptote at a
• lim_{x→a} f(x) does not exist

Correct Answer: lim_{x→a} f(x) exists and equals f(a)

Q39. A vertical asymptote for rational f occurs when:

• Denominator is zero and numerator nonzero at that x
• Numerator and denominator are both zero
• Function approaches a finite value at infinity
• Function is continuous everywhere

Correct Answer: Denominator is zero and numerator nonzero at that x

Q40. A horizontal asymptote y=L indicates:

• lim_{x→a} f(x)=L for finite a
• lim_{x→±∞} f(x)=L
• Function has period L
• Function crosses y=L infinitely often

Correct Answer: lim_{x→±∞} f(x)=L

Q41. Is the absolute value function f(x)=|x| differentiable at x=0?

• Yes, derivative equals 0
• No, not differentiable at 0 due to cusp
• Yes, derivative is 1
• Yes, derivative is −1

Correct Answer: No, not differentiable at 0 due to cusp

Q42. A root (zero) of a function f is:

• An x where f(x)=0
• A point where f is undefined
• A point where derivative is zero only
• A maximum point

Correct Answer: An x where f(x)=0

Q43. For quadratic ax^2+bx+c, the discriminant Δ determines roots. Δ>0 implies:

• Two distinct real roots
• No real roots
• Exactly one real root (double)
• Infinite roots

Correct Answer: Two distinct real roots

Q44. If f'(x) < 0 for all x in interval I, then f is:

• Increasing on I
• Decreasing on I
• Constant on I
• Unbounded on I

Correct Answer: Decreasing on I

Q45. Rolle’s theorem requires which of the following on [a,b]?

• Function continuous on (a,b) and differentiable at endpoints
• Function continuous on [a,b], differentiable on (a,b), and f(a)=f(b)
• Function differentiable on [a,b] only
• Function integrable on [a,b]

Correct Answer: Function continuous on [a,b], differentiable on (a,b), and f(a)=f(b)

Q46. The Mean Value Theorem guarantees c in (a,b) such that:

• f'(c) = (f(b)+f(a))/(b+a)
• f'(c) = f(b)-f(a)
• f'(c) = (f(b)-f(a))/(b-a)
• f'(c) = 0 only if f(a)=f(b)

Correct Answer: f'(c) = (f(b)-f(a))/(b-a)

Q47. A twice-differentiable function with f”(x) ≥ 0 on an interval is called:

• Concave
• Convex (or convex upward)
• Oscillatory
• Periodic

Correct Answer: Convex (or convex upward)

Q48. If f”(x) ≤ 0 on an interval, the function is:

• Convex
• Concave
• Linear
• Undefined

Correct Answer: Concave

Q49. Which statement is true about Lipschitz continuity?

• Lipschitz continuity implies uniform continuity
• Uniform continuity implies Lipschitz continuity always
• Lipschitz functions are never continuous
• Lipschitz continuity is weaker than pointwise continuity

Correct Answer: Lipschitz continuity implies uniform continuity

Q50. In first-order pharmacokinetics with C(t)=C0 e^{-kt}, the half-life t1/2 equals:

• k/ln2
• ln(2)/k
• C0/k
• ln(k)/2

Correct Answer: ln(2)/k

Download