Practice question · Sort into groups
Sort each function by its symmetry.
Groups: Even · Odd · Neither
- x cubed
- x cubed - x
- x squared + x
- x squared
- cos x
- x to the fourth - 3 x squared
Hints
- Replace x by -x and compare the result with the original and with its negative.
- A polynomial with only even powers is even; only odd powers, odd; a mixture, neither.
Show the answer
Even: x squared, x to the fourth - 3 x squared, cos x
Odd: x cubed, x cubed - x
Neither: x squared + x
Why
Even means f(-x) = f(x), giving symmetry about the vertical axis; odd means f(-x) = -f(x), giving symmetry about the origin. Mixing even and odd powers, as in x squared plus x, destroys both, so most functions are neither, and symmetry is a bonus rather than a rule.
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