Question

I would like to work through how to determine whether this is even, odd or netiher.

Answer 1

We are given a function and are asked to determine if it's even, odd or neither. Our approach is to employ the negative test and modify the function to ascertain its status. This will be depicted below.

`h(x)=-6x^3+x^2+8x+8`

This is our original function, next, we find h(-x)

`\begin{gathered} h(x)=-6x^3+x^2+8x+8 \\ h(-x)=-6(-_{}x)^3+(-x)^2+8(-x)+8 \\ h(-x)=6x^3+x^2-8x+8 \\ h(-x)=6x^3+x^2-8x+8 \\ h(-x)=-(-6x^3-x^2+8x-8) \end{gathered}`

Looking at the function h(x) and h(-x), we have to put the polynomial in the correct degree. The input of the negative sign before the h(-x) function is to check if the function will maintain the same format with the original function, h(x). The disparity is what tells us if it is even, odd or neither of the two.

Now we know h(-x), we now use a certain set of conditions to test if function is even, odd or neither.

`\begin{gathered} \text{if h(x)=h(-x), then function is even} \\ \text{if h(-x)=-h(x), then function is odd} \\ \text{if it is neither of the above, function is neither even nor odd.} \end{gathered}`

In this case, based on the above criteria, the function is neither even nor odd.