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MathsMediumClass 11

Sets Functions — Domain Range Composition

Sets Relations Functions

38

JEE Qs

10%

Hard

75

min

Systematically identify and apply all domain restrictions, especially for composite functions, before attempting to determine the range.

🧮 Key Formulas

(f o g)(x) = f(g(x))
Domain(f +/- g) = Domain(f) intersect Domain(g)
Domain(f * g) = Domain(f) intersect Domain(g)
Domain(f / g) = Domain(f) intersect Domain(g) intersect {x | g(x) != 0}
If y = sqrt(f(x)), then f(x) >= 0
If y = 1/f(x), then f(x) != 0
If y = log_b(f(x)), then f(x) > 0, b > 0, b != 1
If y = sin^-1(f(x)) or y = cos^-1(f(x)), then -1 <= f(x) <= 1
Domain(f o g) = {x in Domain(g) | g(x) in Domain(f)}

✅ Key Points for JEE

  • 1To find the domain, identify all restrictions: denominator not zero, argument of square root non-negative, argument of logarithm positive, argument of inverse sine/cosine between -1 and 1.
  • 2For a composite function (f o g)(x), first find the domain of g(x), then ensure that the range of g(x) is a valid subset of the domain of f(x).
  • 3Range can be found by (a) analyzing graphs, (b) expressing x in terms of y (if invertible) and finding domain of inverse, or (c) using calculus (monotonicity, extrema) on the function's domain.
  • 4Always determine the domain of the function before attempting to find its range, as the domain limits the possible output values.
  • 5Memorize the standard domains and ranges of common elementary functions (polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric).

⚠️ Common Mistakes

  • ✕Failing to consider all domain restrictions simultaneously, especially when multiple conditions apply (e.g., nested functions or functions with both square roots and denominators).
  • ✕Incorrectly determining the domain of a composite function (f o g)(x) by only considering the domain of g(x) or only the domain of f(x) without linking them.
  • ✕Finding the range without considering the actual domain of the function, leading to a broader range than correct (e.g., range of x^2 for x in [1,2] is not [0,inf)).

📝 Practice Questions

See all

Q11.Let f(x) = loge x and g(x) = x4−2x3+3x2−2x+22x2−2x+1 . Then the domain of (1) [0, ∞) (2) [1, ∞) (3) (0, ∞) (4) R

2025·MCQMedium

Q22.Let A = {1, 2, 3}. The number of relations on A , containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is ______ -

2025·NumericalMedium

Q17.The number of non-empty equivalence relations on the set {1, 2, 3} is : (1) 6 (2) 5 (3) 7 (4) 4

2025·MCQMedium

Q19.Let A = {1, 2, 3, … , 10} and B = { mn : m, n ∈A, m < n and gcd(m, n) = 1}. Then n(B) is equal to : (1) 36 (2) 31 (3) 37 (4) 29 2025 (22 Jan Shift 1) JEE Main Previous Year Paper

2025·MCQMedium

Q12.Let A = {1, 2, 3, 4} and B = {1, 4, 9, 16}. Then the number of many-one functions f : A →B such that 1 ∈f( A) is equal to : (1) 151 (2) 139 (3) 163 (4) 127

2025·MCQMedium

Q14.Let R = {(1, 2), (2, 3), (3, 3)} be a relation defined on the set {1, 2, 3, 4}. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is: (1) 10 (2) 7 (3) 8 (4) 9

2025·MCQMedium

NCERT Chapters

  • Class 11 Maths Ch 2: Relations and Functions