<s>[INST] For what values of a ​​the function $y=x^6+ax^3-2x^3-2x^2+1$ is even I want to know for what valuyes this function is even
I know that $f(x)=f(-x)$ to proove that function is even. how its helps me?$$y=x^6+ax^3-2x^3-2x^2+1$$
Thanks!
 [/INST] Having even values is different to being an even function. For example, the function $\operatorname{f} : x \mapsto x$ has many even values, e.g. $2 \mapsto 2$ and $4 \mapsto 4$. However, $\operatorname{f}$ is not an even function since $\operatorname{f}(-x) \neq x$ for all $x \neq 0$.
In the example you give, the function cannot be even if it contains odd powers of $x$. Consider 
\begin{array}{ccc}
\operatorname{f}(-x) &\equiv& (-x)^6 + a(-x)^3 - 2(-x)^3 - 2(-x)^2 + 1 \\
&\equiv& (-x)^6 + (a-2)(-x)^3 - 2(-x)^2 + 1 \\
&\equiv& x^6 +(2-a)x^3 -2x^2 + 1
\end{array}
Hence, $\operatorname{f}(x) \equiv \operatorname{f}(-x)$ if and only if $a-2=2-a$, i.e. $a=2$. Notice that $a=2$ if and only if the $x^3$, i.e. the odd powered term, vanishes.
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