Question

Is it possible for an odd function to have the interval 0 infinity) as its domain?

Answer 1

Answer with Step-by-step explanation:

Yes, it is possible for an odd function to have the interval 0 to infinity as its domain.

Let f(x) = -x

f(-x)= -(-x)

     = x

and -f(x)= -(-x)

            = x

Clearly, f(-x) = -f(x)

Hence, f(x) = -x  is an odd function.

and clearly the domain of f(x) = -x is the set of all real numbers which also contains the interval (0,∞)

Hence, it is possible for an odd function to have the interval 0 to infinity as its domain.

Answer 2

By reductio ad absurdum,  it is not possible for an odd function to have the interval $(0, + \infty)$ as its domain.

The domain of a function is the set of x-values so that $f(x)$ exists for all $x$. A function is an odd function if and only if $f(-x) = -f(x)$. We proceed to prove the statement by the reductio ad absurdum approach.

Let suppose that $f(x)$ is an odd function and whose domain is $(0, +\infty)$. If $f(x)$ is odd, then $f(-x) = -f(x)$, which means that function must exist for all $x < 0$, representing an absurd.

Hence, it is not possible for an odd function to have the interval $(0, + \infty)$ as its domain.

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