We are given a function that is the addition of two power functions:
g(x) = x^9 + x^5
Then, in order to find if the function is odd, we need to verify the following identity: g(-x) = - g(x)
Then, first investigate what is "- g(x)" = - ( x^9 + x^5) = - x^9 - x^5
we see that the negative sign distributed into both terms.
Now, let's calculate what g(-x) is to decide if these two expressions are equal:
g(-x) = (-x)^9 + (-x)^5
since -x is equivalent to (-1) * x, we use this coupled with the properties of power of a product to obtain:
(-x)^9 + (-x)^5 = (-1)^9 x^9 + (-1)^5 x^5. Since both, 9 and 5 are ODD numbers, the raising of negative one to such powers will render in both cases "-1":
Then: g(-x) = - x^9 - x^5 which we see is IDENTICAL to the expression we got for "- g(x)"
Therefore, this function is ODD.
It cannot be even, because g(-x) is NOT equal to g(x) (we just proved it is equal to the negative of it.
Therefore, please select the answer that states that g(x) is an ODD function and NOT an even function (first answer option).