Question

How do you know if `F(x) = (2x) / absx` is an even or odd function?

Answer 1

`F` is odd.

Explanation:

Method 1

Consider `F(-x)`,

if `F(-x)=F(x)` for all `x` in the domain of `F`, then `F` is even

if `F(-x)=-F(x)` for all `x` in the domain of `F`, then `F` is odd

if `F(-x)` is not always `F(x)` and not always `-F(x)`, then `F` is neither even nor odd.

`F(x)=(2x)/absx`

`F(-x) = (2(-x))/abs(-x) = (-2x)/absx` `" "` `" "` (Note: `abs(-x)=absx`).

So, `F(-x)=-F(x)` for all `x` in the domain of `F`, and, therefore, `F` is odd.

Method 2

Simplify `F(x)=(2x)/absx` using the definition of absolute value.

`absx = {(" "x," if",x >= 0),(-x," if",x<0) :}`

Note that `F` is not defined at `x=0`, so we get:

`F(x) = {((2x)/x," if",x > 0),((2x)/(-x)," if",x<0) :}`.

Simplifying, gets us:

`F(x) = {(" "2," if",x > 0),(-2," if",x<0) :}`

At this point it is clear that `F(-x) = -F(x)` for all `x` in the domain of `F`. (Because for every `x != 0`, we have `-x` is on the oppposite side of `0`.)