Even and Odd Functions

Parity tells you a graph's symmetry before you draw it. Test even and odd functions, use the parity rules for sums, products and composites, split any function into even and odd parts, and extend a function evenly or oddly.

Builds on: Part 12 · Trigonometric Functions: Graphs and Ranges.

Video coming soon

Definitions

Mirror or rotate

Even: f(−x) = f(x), symmetric about the y-axis (x²). Odd: f(−x) = −f(x), symmetric about the origin (x³). The domain must be symmetric about 0.

Even and odd graphs. — Odd and defined at 0 ⇒ f(0) = 0.

Parity table. — Compare f(−x) with f(x) and −f(x).

Testing

Replace x by −x

x² + cos x even; x³ + sin x odd; x² + x neither; log((1 + x)/(1 − x)) odd; x² on [−1, 2] neither (domain not symmetric); 0 is both.

Decomposition

Even part + odd part

$f (x) = \frac{f ( x ) + f ( - x )}{2} + \frac{f ( x ) - f ( - x )}{2}$ . For eˣ the parts are (eˣ + e⁻ˣ)/2 and (eˣ − e⁻ˣ)/2.

eˣ and its two parts. — They add back to eˣ.

Extensions

From [0, a] to [−a, a]

Even extension f(x) = f(−x), odd extension f(x) = −f(−x) for x in [−a, 0). For x² − x on [0, 2]: x² + x (even) and −x² − x (odd).

Summary

Key formulas

f (x) = \frac{f ( x ) + f ( - x )}{2} + \frac{f ( x ) - f ( - x )}{2}

Even and odd functions

Worked examples

From the video

1. Classify x² + cos x, x³ + sin x, x² + x.

Even, odd, neither.

2. Even and odd extensions of x² − x from [0, 2].

x² + x and −x² − x on [−2, 0).

JEE-style question

Your turn

f(x) = x(eˣ − 1)/(eˣ + 1) is:

(a)odd

(b)even

(c)neither even nor odd

(d)both even and odd

Show the answer and the traps

(eˣ − 1)/(eˣ + 1) is odd: multiply top and bottom of g(−x) by eˣ. Times the odd x, it gives an even product. Option (b).

(a) misses that odd × odd is even. (c) gives up because eˣ itself is neither.

(d) would need f to be the zero function.

Watch out

Common mistakes

Calling a function even or odd on a non-symmetric domainThen it is neither.

Forgetting f(x) = 0 is bothIt is the only function that is both on a given domain.

Assuming odd × odd is oddProducts follow sign rules: odd × odd is even.

Practice

Try these

1. Is log(x + √(x² + 1)) even or odd?

Odd.

2. Even and odd parts of x² + 3x + 2.

x² + 2 and 3x.

3. Is (2ˣ + 1)/(2ˣ − 1) even or odd?

Odd.