Is f(x)=x^2+sin x an even or odd function?

2 Answers
Jim H
Sep 26, 2015

It is neither.

Explanation:

A function f is even if and only if f(-x)=f(x) for every x in the domain of f.
A function f is odd if and only if f(-x)=-f(x) for every x in the domain of f.

For f(x)=x^2+sin x, we check f(-x).

If f(-x) simplifies to f(x), then f is even,
If f(-x) simplifies to -f(x) =-(x^2+sin x), the f is odd.
If f(-x) does not simplify to one of the above, then f is neither even nor odd.

f(-x) = (-x)^2+sin(-x)

= x^2-sinx which is neither f(x) nor -f(x)

We cannot show that f(-x)=f(x) for every x by showing that it is true for only one x,
but we can show that is fails to be true for all x by showing that it fails for one value.

f(pi/2)=pi^2/4+1
f(-pi/2) =pi^2/4-1

It is, I think, clear that these numbers are neither equal nor negatives of each other.

George C.
Sep 26, 2015

Neither.

f(-pi/6) = pi^2/36 - 1/2 ~~ 10/36 - 1/2 = -2/9

f(pi/6) = pi^2/36 + 1/2 ~~ 10/36 + 1/2 = 7/9

So f(-pi/6) != +-f(pi/6)

Explanation:

An even function requires f(-x) = f(x)

An odd function requires f(-x) = -f(x)

Our example satisfies neither of these conditions.

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