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MathsMediumClass 11
Domain & Range — Finding for various functions
Sets, Relations & Functions
10
JEE Qs
8%
Hard
75
min
Thoroughly identify all restrictions for the domain using set intersection, and for the range, employ algebraic manipulation, graphical interpretation, or properties of standard functions.
🧮 Key Formulas
Domain of sqrt(f(x)) is where f(x) >= 0
Domain of 1/f(x) is where f(x) != 0
Domain of 1/sqrt(f(x)) is where f(x) > 0
Domain of log_b(a) is where a > 0, b > 0, b != 1
Domain of sin^-1(x) and cos^-1(x) is [-1, 1]
Domain of tan^-1(x) and cot^-1(x) is (-infinity, infinity)
Domain of sec^-1(x) and csc^-1(x) is (-infinity, -1] U [1, infinity)
✅ Key Points for JEE
- 1Domain is the set of all real numbers for which the function f(x) is well-defined and yields a real value.
- 2When multiple restrictions apply (e.g., root and denominator), the domain is the intersection of all individual valid intervals.
- 3For finding the range, consider algebraic manipulation (making x the subject), graphical analysis, properties of basic functions (like min/max values of quadratics, trig functions), or calculus (differentiation for extrema).
- 4For composite functions f(g(x)), the domain is determined by conditions on x such that g(x) is in the domain of f(x).
- 5Always check for implicit restrictions, such as the natural domain of a function even if not explicitly stated (e.g., tan(x) is undefined at odd multiples of pi/2).
⚠️ Common Mistakes
- ✕Forgetting to combine all domain restrictions using intersection; e.g., for 1/sqrt(f(x)), correctly applying f(x) > 0 instead of f(x) >= 0.
- ✕Incorrectly solving inequalities, especially those involving rational expressions, absolute values, or multiple factors, leading to wrong intervals.
- ✕Assuming the range of a function directly from its form without considering the domain or using proper analytical methods (e.g., confusing range of x^2 with range of (x+1)^2+2).
- ✕Not considering the range of the inner function when finding the domain of a composite function; the range of the inner function must lie within the domain of the outer function.
NCERT Chapters
- Class 11 Maths Ch 2: Relations and Functions
- Class 11 Maths Ch 6: Linear Inequalities
- Class 12 Maths Ch 2: Inverse Trigonometric Functions