Question

How do you determine if `f(x)= 1+ sinx` is an even or odd function?

Answer 1

This function is neither even nor odd, since neither `f(-x) = f(x)` nor `f(-x) = -f(x)` holds.

Explanation:

An even function is one that satisfies `f(-x) = f(x)` for all `x` in its domain.

An odd function is one that satisfies `f(-x) = -f(x)` for all `x` in its domain.

`f(x) = 1 + sin(x)`

is the sum of a non-zero even function `1` and a non-zero odd function `sin(x)`.

As a result it is neither even nor odd.

For example:

`f(-pi/2) = 1 + sin(-pi/2) = 1 - 1 = 0`

`f(pi/2) = 1 + sin(pi/2) = 1 + 1 = 2`

So we have neither `f(-x) = f(x)` nor `f(-x) = -f(x)`