Even, Odd and Periodic Functions
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Direct answer
Before differentiating, integrating or sketching any function, interrogate its symmetry: even means f(−x) = f(x), a mirror in the y-axis (x², cos x); odd means f(−x) = −f(x), a half-turn about the origin (x³, sin x); periodic means f(x + T) = f(x) for fixed positive T, the smallest such T the period — 2π for sine and cosine, π for tangent. The tests are single substitutions, and the payoffs are computational: an odd function integrates to zero over [−a, a], even functions halve the work, periodic functions repeat every calculation. Products follow sign rules — odd × odd = even, odd × even = odd — and a polynomial's parity is decided by which powers survive.
What you must remember
- The tests: substitute −x once: same expression means even, negated means odd, neither means neither (most functions are neither; x² + x is the canonical witness).
- Standard periods: sin, cos, sec, cosec carry 2π; tan, cot carry π; |sin x| and sin²x, cos²x carry π; the fractional part {x} carries 1; [x] is not periodic at all.
- Halving by squaring: |sin x| and sin²x halve the period because both lift the negative half-cycle — but |sin x| + sin x keeps 2π, since the plain sine term still needs its full round.
- Sums and LCM logic: the sum of functions with periods T1 and T2 is periodic when T1/T2 is rational, with period dividing the LCM; sin⁴x + cos⁴x = 1 − ½sin²2x has period π/2.
- Composition: a periodic input inside anything stays periodic — sin(anything periodic) is periodic; sin(x²) is not, since squaring destroys the shift structure.
- Calculus mirrors: the derivative of an even function is odd and vice versa; the integral from 0 to x of an odd function is even.
- Antiperiodic nuance: if f(x + T) = −f(x), the period is 2T — ln((1 + sin x)/(1 − sin x)) shows this with T = π.
One function, three tests
Classify f(x) = ln((1 + sin x)/(1 − sin x)) completely. Parity: f(−x) = ln((1 − sin x)/(1 + sin x)) = −ln((1 + sin x)/(1 − sin x)) = −f(x), so the function is odd — the logarithm of a reciprocal negates. Periodicity: f(x + π) replaces sin x by −sin x, which flips the fraction and negates the value, so f(x + π) = −f(x) — antiperiodic behaviour, meaning the true period is 2π, not π. Domain: sin x = ±1 must be excluded, so x ≠ π/2 + kπ punctures the line. Three properties, three single substitutions, and the payoff: knowing f is odd means any integral over a symmetric interval dies immediately; knowing the period 2π means any property computed on one round transfers everywhere. This is the entire method — substitution before intuition, because graphs mislead exactly when formulas do not, and the antiperiodic half-step is invisible to casual sketching.
Period pitfalls
JEE Main asks identification from formulas and from graphs, with the planted distractors being π for anything containing |sin x| blindly (wrong when a 2π-periodic partner rides along) and 2π for sin²x (wrong in the other direction — the square already lifted the graph). The LCM questions carry rational-ratio checks: sin x + sin(π√2 x) is not periodic at all, since the period ratio is irrational, and that non-periodicity is itself a favourite option. JEE Advanced exploits the integral savings: an odd integrand on a symmetric interval is zero without computation, and half the work of a periodic integrand over whole periods collapses by periodicity — the king-property and symmetry tools of definite integration are this chapter wearing calculus clothes. The composition trap deserves a drill: sin(x²) and sin(1/x) are not periodic no matter how periodic sine itself is, because the argument's advance does not shift the function by a constant. Rehearse the substitutions until they are reflexes; every property here costs one line and buys whole questions.
Frequently asked questions
How do you test whether a function is even, odd or neither?
Substitute −x: an identical result means even, a negated result means odd, and anything else means neither.
What is the period of |sin x| and of sin²x?
π for both — taking the absolute value or the square lifts the negative half-cycle, halving sine's period.
When is a sum of periodic functions itself periodic?
When the ratio of the individual periods is rational; the combined period then divides the LCM of the periods.
Why does f(x + T) = −f(x) imply period 2T?
Applying the shift twice returns the original value: f(x + 2T) = −f(x + T) = f(x), and 2T is typically the smallest such shift.
How does symmetry save work in integration?
An odd integrand integrates to zero over [−a, a]; an even one gives double the half-interval integral — answers without antiderivatives.