These questions give me a problem since the rules of distribution seem not to apply to them, for example:
Determine whether the following function is even, odd, or neither?
$g(x)= 1-x^4 $
$g(-x) =1 – (-1x)^4$
$1-(+1x^4)$
distribute the negative
$1-1x^4$
factor out the negative
$-1 (-1 +1x^4)$
Thus this function is neither, however, the book says it is even, why?
Why are the rules of distribution of the negative sign not being applied here?
Best Answer
Recall that $(-x)^2 = (-x)(-x) = x^2\geq 0 \forall x$. And note that $(-x)^4 = ((-x)^2)^2 = (x^2)^2 = x^4$.
Indeed, $(-x)^{2n} = x^{2n}$ when $n \in \mathbb Z$.
$$g(-x) = (1-(-x)^4) = 1 - x^4 = g(x) $$
which is what you found after "distributing the negative": $1 - 1\cdot x^4 = 1- x^4$.
So, what does it mean when $g(x) = g(-x)?$