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Questions on even, odd and periodic functions are short, but they reward a habit most students skip: simplify the function before applying a rule. The LCM rule for periods, in particular, gives a wrong answer in some common cases. This post shows how to test for even and odd, how to find periods, where the rules break, and why these properties save time later in integration.
Where is this in the syllabus?
The JEE (Advanced) 2026 brochure lists "even and odd functions" among the topics on functions, and "Trigonometric functions, their periodicity and graphs" under trigonometry. The JEE (Main) 2026 syllabus lists real-valued functions, algebra of functions and "Graphs of simple functions" without naming these properties. NCERT's Class 12 chapter on integrals (2026-27 reprint) uses even and odd functions in a property of definite integrals. Check your exam's current documents.
How do you test whether a function is even or odd?
Replace x by −x and simplify.
Even: f(−x) = f(x). The graph is symmetric about the y-axis. Examples: x², cos x, |x|.
Odd: f(−x) = −f(x). The graph is symmetric about the origin. Examples: x³, sin x, tan x.
Products: even × even and odd × odd are even; even × odd is odd.
The domain must be symmetric. x² on [−1, 2] is neither, because f(2) exists and f(−2) does not.
Any function on a symmetric domain is the sum of an even part, [f(x) + f(−x)]/2, and an odd part, [f(x) − f(−x)]/2. For eˣ, these are (eˣ + e⁻ˣ)/2 and (eˣ − e⁻ˣ)/2.
Worked examples: even or odd?
Example 1. f(x) = log(x + √(x² + 1)).
f(−x) = log(−x + √(x² + 1)). Multiply and divide by x + √(x² + 1): the numerator becomes (x² + 1) − x² = 1, so f(−x) = log(1/(x + √(x² + 1))) = −f(x). The function is odd.
Example 2. g(x) = x/(eˣ − 1) + x/2, for x ≠ 0.
g(−x) = −x/(e⁻ˣ − 1) − x/2. Multiply the first fraction's top and bottom by eˣ: −xeˣ/(1 − eˣ) = xeˣ/(eˣ − 1). Then g(x) − g(−x) = [x − xeˣ]/(eˣ − 1) + x = −x + x = 0. The function is even, although neither part looks it.
Why it pays off. NCERT's property P7 states that the integral from −a to a of an odd function is 0, and of an even function is twice the integral from 0 to a. So the integral of x³ cos x from −1 to 1 is 0 at sight, because x³ is odd and cos x is even. Our post on properties of definite integrals has more.
How do you find the period?
The period is the smallest positive T with f(x + T) = f(x) for every x in the domain.
sin(ax + b) and cos(ax + b): 2π/|a|.
tan(ax + b): π/|a|.
|sin x| and |cos x|: π. The fractional part {x}: 1.
For sums, take the LCM. sin 2x + cos 3x has periods π and 2π/3. The smallest number that is a whole multiple of both is 2π. For tan 3x + sin(x/2), the periods are π/3 and 4π, and the LCM is 4π.
When does the LCM rule give the wrong answer?
sin⁴x + cos⁴x. Each term has period π, but sin⁴x + cos⁴x = 1 − (1/2)sin²2x = 3/4 + (1/4)cos 4x, so the period is π/2.
|sin x| + |cos x|. Each term has period π, but shifting by π/2 swaps the two terms and leaves the sum unchanged, so the period is π/2.
No common multiple. sin x + {x} has periods 2π and 1. No whole multiple of 2π is a whole number, because π is irrational, so the sum is not periodic.
Not periodic at all. sin(x²) and x + sin x repeat no pattern, whatever you try.
Which mistakes cost the most marks?
Taking the LCM without simplifying.
Calling a function even after testing one value, such as f(1) = f(−1).
Ignoring the domain when testing for even or odd.
Forgetting |a| in 2π/|a|, giving a negative period.
A practice routine
Knowing why a rule works is what tells you when it fails. In two experiments, Rittle-Johnson, Siegler and Alibali (2001) found that conceptual and procedural knowledge "develop iteratively", each predicting gains in the other.
Simplify first. For every period question, write the simplified form before any LCM.
Test half the answer every time.
Sketch symmetry. For even and odd questions, sketch the graph and check its symmetry. Our post on graph transformations helps here.
Mix with integrals. Add two definite integrals from −a to a to each set.
If these graphs are hard to picture, the Fermi Simulation Lab has interactive simulations for ideas easier to see than read. For the step functions that complicate periods, see our post on greatest integer and fractional part functions.
Today, find the period of sin⁶x + cos⁶x, then check whether half your answer also works.
