Functions - Odd and Even Functions
Functions - Odd and Even Functions
Even and Odd Functions
A function \(f\) is even if \(f(-x)=f(x)\) for all \(x\) in its domain.
A function \(f\) is odd if \(f(-x)=-f(x)\) for all \(x\) in its domain.
A function is neither even nor odd if \(f(-x)\ne f(x)\) and \(f(-x)\ne -f(x)\).
| Type of function | Algebraic condition | Graphical meaning |
|---|---|---|
| Even | \(f(-x)=f(x)\) | Symmetrical about the \(y\)-axis |
| Odd | \(f(-x)=-f(x)\) | Rotational symmetry of \(180^\circ\) about the origin |
| Neither | \(f(-x)\ne f(x)\) and \(f(-x)\ne -f(x)\) | No required symmetry |
To test whether a function is even, odd, or neither, first find \(f(-x)\). Then compare it with \(f(x)\) and \(-f(x)\).
- If \(f(-x)=f(x)\), then \(f\) is even.
- If \(f(-x)=-f(x)\), then \(f\) is odd.
- If neither condition is true, then \(f\) is neither even nor odd.
Important IB HL point: the domain must be symmetrical about \(0\). For example, \([-3,3]\) is symmetrical about \(0\), but \([0,3]\) is not.
| Function | Type |
|---|---|
| \(x^2\) | Even |
| \(x^4-3x^2+1\) | Even |
| \(\cos x\) | Even |
| \(x^3\) | Odd |
| \(x^5-2x\) | Odd |
| \(\sin x\) | Odd |
| \(x^2+x\) | Neither |
| \(x^3+1\) | Neither |
Useful IB HL calculus results:
- If \(f\) is even, then \(\int_{-a}^{a} f(x)\,dx=2\int_0^a f(x)\,dx\).
- If \(f\) is odd, then \(\int_{-a}^{a} f(x)\,dx=0\).
Common mistake: a function with odd powers plus a constant is usually neither odd nor even. For example, \(f(x)=x^3+1\) is neither even nor odd.
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