# Even, Odd and Periodic Functions

> Even, odd and periodic functions for JEE Mathematics: substitution tests, standard periods, LCM logic, antiperiodic cases and integral savings.

- Canonical URL: https://prepelephant.com/topics/jee/mathematics/even-odd-periodic-identification
- Exam / course: JEE · Subject: Mathematics
- Publisher: PrepElephant (https://prepelephant.com) — Prepared and reviewed by the PrepElephant Academic Review Team
- First published: 2026-10-02
- Last updated: 2026-10-02
- How to cite: "Even, Odd and Periodic Functions", PrepElephant, https://prepelephant.com/topics/jee/mathematics/even-odd-periodic-identification

## 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.