6.5 Even and Odd functions
A function \(f\) is called an even function if \(f(-x)=f(x)\).
e.g the function \(f(x)=x^2\) is even since \(f(-x)=(-x)^2=x^2=f(x)\).
A function \(f\) is called an odd function if \(f(-x)=-f(x)\).
e.g \(f(x)=x^3+x\) is odd. Since \begin {align*} f(-x) & = (-x)^3+(-x)=-x^3-x\\ & = -(x^3+x)\\ & = -f(x)\\ \end {align*}
Determine whether each of the following functions is even, odd or neither
- 1.
- \(f(x)=3x^4-2x^2\)
- 2.
- \(f(x)=x^3+x-2\)
- 3.
- \(f(x)=2x^3-\frac {1}{x}\)
Solution.
In each case compute \(f(-x)\) and compare it with \(f(x)\) and with \(-f(x)\).
1. For \(f(x)=3x^{4}-2x^{2}\), \begin {align*} f(-x) & = 3(-x)^{4}-2(-x)^{2}\\ & = 3x^{4}-2x^{2}\\ & = f(x), \end {align*}
so \(f\) is even. Both powers are even, which is what makes the signs cancel.
2. For \(f(x)=x^{3}+x-2\), \begin {align*} f(-x) & = (-x)^{3}+(-x)-2\\ & = -x^{3}-x-2 . \end {align*}
Compare this with the two things it would have to equal: \[f(x)=x^{3}+x-2,\qquad -f(x)=-x^{3}-x+2 .\] It matches neither — the constant term has the wrong sign in each case — so \(f\) is neither even nor odd.
3. For \(f(x)=2x^{3}-\dfrac {1}{x}\), \begin {align*} f(-x) & = 2(-x)^{3}-\frac {1}{-x}\\ & = -2x^{3}+\frac {1}{x}\\ & = -\left (2x^{3}-\frac {1}{x}\right )\\ & = -f(x), \end {align*}
so \(f\) is odd.
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