Is this alternative odd function equation valid?

Question

An odd function is a function where:

$$f(-x) = -f(x)$$

Is this alternative form valid ?

$$f(x) = -f(-x)$$

I'm asking because an odd function graphically is symmetrical with respects to the origin. Which means that the x and y point's signs are flipped.

The second form seems easier to read (assuming it's valid).

Answer 1

(CW; just ensuring this question registers as answered.)

Yes: As remarked in the comments, the two equations are equivalent. This can be seen by simply multiplying both sides of one equation by $-1$ to obtain the other equation.

As to which one "seems easier to read," this is -- of course -- a matter of taste.

Answer 2

Yes, as long as $(-1)^2=1$. You have:

$$f(-x) = -f(x) \Rightarrow (-1)f(-x) = (-1)(-f(x)) \Rightarrow -f(-x) = (-1)(-1)f(x) = f(x) \Rightarrow f(x) = -f(-x)$$

and of course the reverse:

$$f(x) = -f(-x) \Rightarrow (-1)f(x) = (-1)(-1)f(-x) \Rightarrow -f(x) = f(-x) \Rightarrow f(-x) = -f(x)$$

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While what seems easier to read is a matter of taste they generalize differently. If you replace negation with another operation(*) they will no longer be the same - not that it matters in your context, in another context one of the definitions might be more natural than the other.

(*) For example homogenity is a stronger condition which is defined as $cf(x) = f(cx)$ for any constant $c$ (not only $-1$). Here you see it might be more "natural" to define oddness with the first definition.