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Calculus/Even and Odd Functions

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Expert: - 4/13/2009


Question
I was wondering if you could help me out on even and odd functions. My
question is how can you tell if a function is even or odd just by looking at the
equation not the graph. For instance this equation f(x)=x^3-x+4

Answer
For an even function, the exponets must all be even.
for an odd function, the exponets must all be odd.

For the equation f(x) = x^3 -x + 4,
almost all of the exponets are odd,
which would make it an odd function.

Note that on 4, it's really 4*x^0 (since x^0 is always 1).
Note that 0 is even, so the function is neither.

If the function were just x^3 - x, it would be odd.

If the function were just 4, it would be even.

If 4 is subtracted from the function, it becomes and odd function.

In proper definitions, an even function is one in which f(x)=f(-x)
and an odd function is one in which f(x)=-f(-x).

For even functions, the y axis is a mirror.

For odd functions, the y axis is a reflective mirror
that produces an upside down image of the graph.

If the y axis were put at y=4 in this problem making the function
f(x) = x^3 - x, that would be an odd function.

Calculus

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