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Determine whether the function below is an even function, an odd function, both, or neither.

[tex]f(x) = x^6 + 10x^4 - 11x^2 + 19[/tex]

A. even function
B. neither even nor odd
C. both even and odd
D. odd function


Sagot :

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