To determine whether the function [tex]\( f(x) = x^6 + 10x^4 - 11x^2 + 19 \)[/tex] is an even function, an odd function, or neither, we need to test the properties of even and odd functions.
### Definitions:
- A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex].
- A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex].
### Step-by-Step Solution:
1. Calculate [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = (-x)^6 + 10(-x)^4 - 11(-x)^2 + 19
\][/tex]
2. Simplify [tex]\( f(-x) \)[/tex]:
[tex]\[
(-x)^6 = x^6
\][/tex]
[tex]\[
10(-x)^4 = 10x^4
\][/tex]
[tex]\[
-11(-x)^2 = -11x^2
\][/tex]
[tex]\[
19 \text{ is constant and does not change with \( x \)}
\][/tex]
Therefore,
[tex]\[
f(-x) = x^6 + 10x^4 - 11x^2 + 19
\][/tex]
3. Compare [tex]\( f(-x) \)[/tex] and [tex]\( f(x) \)[/tex]:
[tex]\[
f(-x) = x^6 + 10x^4 - 11x^2 + 19
\][/tex]
[tex]\[
f(x) = x^6 + 10x^4 - 11x^2 + 19
\][/tex]
We see that [tex]\( f(-x) = f(x) \)[/tex].
### Conclusion:
Since [tex]\( f(-x) = f(x) \)[/tex], the function [tex]\( f(x) = x^6 + 10x^4 - 11x^2 + 19 \)[/tex] is an even function.
Therefore, the correct answer is:
A. even function
