IB Maths AA HLTopic 2 — FunctionsPaper 1 & 2~7 min read
Odd & Even Functions
Some functions have a special kind of symmetry. Even functions look the same on both sides of the y-axis (mirror image). Odd functions look the same after a 180° spin about the origin (rotational). Most functions are neither — but if a function is odd or even, the property often makes integration, sketching, and series questions much faster. The test is one quick algebraic check: substitute −x and see what happens.
📘 What you need to know
Even: f(−x) = f(x) for all x. Graph has reflective symmetry in the y-axis.
Odd: f(−x) = −f(x) for all x. Graph has rotational symmetry (180°) about the origin.
Most functions are neither. If f(−x) doesn’t equal f(x) AND doesn’t equal −f(x), it’s neither.
Even powers (x0, x2, x4, …) are even. Odd powers (x, x3, x5, …) are odd. That’s where the names come from.
Trig: cosx, secx are even. sinx, tanx, cosecx, cotx are odd.
|x| is even. Constants (e.g. f(x) = 7) are even.
Combination rules: even ± even = even; odd ± odd = odd; even × even = even; odd × odd = even; even × odd = odd.
The two definitions
Even — mirror in y-axis
f(−x) = f(x)
e.g. x2, x4, cosx, |x|, constants
Odd — 180° rotation about O
f(−x) = −f(x)
e.g. x, x3, x5, sinx, tanx
Power-rule mnemonic: an integer power xn is even when n is even, and odd when n is odd. Same word, same meaning.
What the symmetry looks like
Even is a mirror; odd is a spin
Quick GDC trick: turn your calculator upside down (180° rotation). If the graph looks the same, it’s odd. Sounds silly but it works.
Combination rules — predict without testing
Once you know the type of each piece, you can often work out the type of the whole expression without algebra:
Sum/difference
even ± even = even odd ± odd = odd
mixing even and odd → usually neither
Product (same type)
even × even = even odd × odd = even
two negatives multiply to positive
Product (mixed)
even × odd = odd
odd “wins” against even
Mnemonic: with multiplication, treat even = +1 and odd = −1, then multiply the signs. Two odds give +1 (even). One of each gives −1 (odd). Easy to remember.
How to test algebraically
🧭 Recipe — testing for odd/even
Compute f(−x) by replacing every x with −x in the expression.
Simplify using rules like (−x)2 = x2, sin(−x) = −sinx, cos(−x) = cosx.
Compare: if it equals f(x) → even. If it equals −f(x) → odd. Otherwise → neither.
Worked examples
WE 1
Test if a polynomial is odd, even, or neither
Show that f(x) = 5x3 − 2x is odd.
Step 1: Compute f(−x)f(−x) = 5(−x)³ − 2(−x)= −5x³ + 2xStep 2: Compare to ±f(x)−f(x) = −(5x³ − 2x) = −5x³ + 2xf(−x) = −f(x) ✓f is oddall terms have odd powers of x → expected to be odd
WE 2
Test a polynomial — even case
Show that f(x) = 2x4 + 3x2 − 1 is even.
Step 1: Compute f(−x)f(−x) = 2(−x)⁴ + 3(−x)² − 1= 2x⁴ + 3x² − 1Step 2: Compare to f(x)f(−x) = f(x) ✓f is evenall terms have even powers (counting the −1 as x⁰) → expected to be even
WE 3
A function that is neither odd nor even
Determine whether f(x) = x3 + 4x2 + x is odd, even, or neither.
Step 1: Compute f(−x)f(−x) = (−x)³ + 4(−x)² + (−x)= −x³ + 4x² − xStep 2: Compare both waysf(x) = x³ + 4x² + x → not equal to f(−x), not even−f(x) = −x³ − 4x² − x → not equal to f(−x), not oddf is neither odd nor evenmixing odd-power terms (x³, x) with an even-power term (4x²) usually gives neither
WE 4
Use combination rules — show even
Show, using combination rules, that g(x) = x2 cos(3x) + 7 is even.
Identify each piecex² is even (even power)cos(3x) is even (cos of any expression in x: cos(−3x) = cos(3x))7 is even (constant)Apply combination rulesx² · cos(3x): even × even = eveneven + even = eveng(x) is evenverify: g(−x) = (−x)²cos(−3x) + 7 = x² cos(3x) + 7 = g(x) ✓
WE 5
Use combination rules — show odd
Show that h(x) = 2x3 − sin(5x) is odd.
Identify each piece2x³ is odd (constant × odd)sin(5x) is odd (sin of an odd-input expression: sin(−5x) = −sin(5x))Apply combination rulesodd − odd = oddVerify directlyh(−x) = 2(−x)³ − sin(−5x) = −2x³ + sin(5x)= −(2x³ − sin(5x)) = −h(x) ✓h(x) is odda multiplicative constant (2 here) doesn’t change parity — only the structure of the variable parts does
WE 6
Identify the type from a graph description
A graph of y = f(x) is symmetric about the y-axis: every point (a, b) on the curve has a partner (−a, b). State whether f is odd, even, or neither, with a reason.
Identify the symmetry typereflection in the y-axis ↔ even(if it had been 180° rotation about origin → odd)f is even, because (a, b) and (−a, b) both lie on the graph, so f(−a) = f(a) = b for all agraphs alone are usually enough — pick any input and check the partner point on the other side
💡 Top tips
Power test for polynomials: all-even powers → even; all-odd powers → odd; mixed → neither.
Trig in one line: cos and sec are even; sin, tan, cot, cosec are odd.
Combination rules save algebra. Identify each piece’s parity, multiply/sum the parities to get the whole.
Constants count as even, including the constant term in a polynomial. So x3 + 5 is neither (odd + even).
Function inside a function: if you have sin(u) where u is an odd expression in x, sin(u) is odd. If u is even, sin(u) is even.
Use the graph test: y-axis mirror → even; 180° rotation about origin → odd.
If f(0) ≠ 0 and f is odd, something’s wrong — every odd function passes through the origin.
⚠ Common mistakes
Forgetting that the constant term breaks the parity.x3 is odd, but x3 + 1 is neither.
Treating cos as odd by analogy with sin. Cos is even — make sure you’ve memorised which is which.
Sign errors with negative powers. (−x)3 = −x3, but (−x)2 = +x2. Watch the parity of the power.
Concluding “neither” when a function is just non-trivially odd. Always factor out a minus sign at the end of f(−x) to compare with −f(x).
Applying combination rules wrong. Even × odd = odd, not even. The classic slip-up.
Forgetting that “neither” is a valid answer. Most polynomials with mixed-parity terms are neither.
Assuming a function with a symmetric graph must be even. Symmetry about the y-axis gives even; symmetry about the origin (rotational) gives odd. Check which kind.
Odd and even functions become especially powerful in calculus — the integral of an odd function over a symmetric interval [−a, a] is automatically zero, and integrating an even function only requires the right half (then double it). The next note shifts to periodic functions: graphs that repeat the same pattern over and over — sin, cos, tan and their relatives.
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