3.4 Odd and even functions
Definition 3.14. A function \(f\) is even if \(f(-x)=f(x)\) for every \(x\), and odd if \(f(-x)=-f(x)\) for every \(x\).
An even function is unchanged when the sign of the input flips; an odd one changes sign with it. Graphically, an even function is symmetric about the \(y\)-axis and an odd one has rotational symmetry about the origin.
Example 3.15. State whether each function is even, odd, or neither.
- (a).
- \(f(x)=\frac {x^3-x}{x^2+4}\)
- (b).
- \(f(x)=x^4-3x^2+7\)
- (c).
- \(f(x)=2x+7\)
Solution. In every case the method is the same: work out \(f(-x)\), then compare it with \(f(x)\) and with \(-f(x)\).
(a) Replace each \(x\) by \(-x\), remembering \((-x)^3=-x^3\) and \((-x)^2=x^2\): \[f(-x)=\frac {(-x)^3-(-x)}{(-x)^2+4}=\frac {-x^3+x}{x^2+4}=-\left (\frac {x^3-x}{x^2+4}\right )=-f(x).\] So \(f\) is odd.
(b) Only even powers appear, and the constant is unaffected: \[f(-x)=(-x)^4-3(-x)^2+7=x^4-3x^2+7=f(x).\] So \(f\) is even.
(c) \[f(-x)=2(-x)+7=-2x+7.\] This is not \(f(x)=2x+7\), so \(f\) is not even. Nor is it \(-f(x)=-2x-7\), since the constants differ. So \(f\) is neither.
Note 3.16. Most functions are neither odd nor even — those two are special cases, not alternatives. Part (c) shows how a single constant term can spoil oddness: \(2x\) on its own is odd, but \(2x+7\) is not.
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