Given:
[tex]\begin{gathered} g(x)=-x^5-4x^3+6x \\ \\ h(x)=x^4+2x^3-2x^2+x-7 \\ \\ j(x)=3x^4+7x^2 \end{gathered}[/tex]
It's required to determine if the functions are odd, even, or neither.
An even function satisfies the property:
f(-x) = f(x).
And an odd function satisfies the property:
f(-x) = -f(x)
We substitute x by -x on each function as follows:
[tex]\begin{gathered} g(-x)=-(-x)^5-4(-x)^3+6(-x) \\ \\ g(-x)=x^5+4x-6x \end{gathered}[/tex]
Note the function g(-x) is the inverse (negative) of g(x), thus,
g(x) is odd
Now test h(x):
[tex]\begin{gathered} h(-x)=(-x)^4+2(-x)^3-2(-x)^2+(-x)-7 \\ \\ h(-x)=x^4-2x^3-2x^2-x-7 \end{gathered}[/tex]
Comparing h(-x) and h(x) we can see none of the properties are satisfied, thus:
h(x) is neither odd nor even
Let's now test j(x):
[tex]\begin{gathered} j(-x)=3(-x)^4+7(-x)^2 \\ \\ j(-x)=3x^4+7x^2 \end{gathered}[/tex]
Since j(-x) and j(x) are equal,
j(x) is even
