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MathsMediumClass 11
Even & Odd Functions
Sets, Relations & Functions
10
JEE Qs
8%
Hard
45
min
Always begin by checking for domain symmetry, then precisely substitute -x into the function and simplify to determine its parity.
🧮 Key Formulas
A function f(x) is even if f(-x) = f(x) for all x in its domain.
A function f(x) is odd if f(-x) = -f(x) for all x in its domain.
Any function f(x) with a symmetric domain can be expressed as the sum of an even and an odd function: f(x) = E(x) + O(x), where E(x) = [f(x) + f(-x)]/2 (even part) and O(x) = [f(x) - f(-x)]/2 (odd part).
✅ Key Points for JEE
- 1For a function to be classified as even or odd, its domain must be symmetric about the origin (i.e., if x is in the domain, then -x must also be in the domain). If the domain is not symmetric, the function is neither even nor odd.
- 2The product/quotient of two even functions is even. The product/quotient of two odd functions is even. The product/quotient of an even and an odd function is odd.
- 3The sum/difference of two even functions is even. The sum/difference of two odd functions is odd. The sum/difference of an even and an odd function is generally neither, unless one of the functions is identically zero.
- 4The only function that is both even and odd is f(x) = 0 for all x in its domain.
- 5Graphically, an even function is symmetric about the y-axis, and an odd function is symmetric about the origin.
- 6The derivative of an even function is odd, and the derivative of an odd function is even (assuming differentiability). For definite integrals with symmetric limits, ∫[-a,a] f(x) dx = 2∫[0,a] f(x) dx if f is even, and ∫[-a,a] f(x) dx = 0 if f is odd.
⚠️ Common Mistakes
- ✕Assuming every function is either even or odd; many functions are neither.
- ✕Forgetting to first check if the domain of the function is symmetric about the origin before applying the even/odd test.
- ✕Making algebraic errors when substituting -x into the function definition, especially with complex expressions or functions involving absolute values and piecewise definitions.
NCERT Chapters
- Class 11 Maths Ch 2: Relations and Functions