Even And Odd Functions

An odd function satisfies f(-x) = -f(x), giving symmetry about the origin, a definition repeatedly exploited in JEE Advanced integration and graph problems. Test f(x) = x^3 + sin x: f(-x) = (-x)^3 + sin(-x) = -x^3 - sin x = -(x^3 + sin x) = -f(x), so it is odd. The sum of two odd functions, x^3 and sin x, remains odd, illustrating the closure of odd functions under addition. Option x^2 + cos x is even because both terms are even. Option x^3 + cos x mixes an odd term with an even term, so it has no definite parity. Option e^x + x is neither, since e^x has no symmetry. Hence the odd function is x^3 + sin x. Plausibility check: evaluating at x = 1 and x = -1 gives f(1) = 1 + sin 1 and f(-1) = -1 - sin 1 = -f(1), numerically confirming the origin symmetry expected of an odd function. Every real function splits uniquely into an even component (f(x)+f(-x))/2 and an odd component (f(x)-f(-x))/2, so parity questions reduce to inspecting these parts. This decomposition powers many definite-integral shortcuts in which the odd component integrates to zero over a symmetric interval, linking the parity idea directly to calculus techniques tested later in the syllabus.