Question

Prove algebraically what type of function this is (even, odd, or neither).
f(x) = x^6 – x^4​

Answer 1

Let's see.

We have function $f(x)=x^6-x^4$. We have to determine if the function is even or odd or neither.

We can find this out in many ways but the assignment specifically asks to prove this algebraically.

Function is even if $f(x)=f(-x)$, so, $x^6-x^4=(-x)^6-(-x)^4$

Which is true since negative raised to an even power is positive to that same number.

Similarly, function is odd if $f(x)=-f(-x)$, that is,

$x^6-x^4\neq-((-x)^6-(-x)^4)$.

So we have prooved that function $f(x)$ is an even function not an odd function and therefore also not neither.

Hope this helps.