Find solutions to your questions with the help of IDNLearn.com's expert community. Find reliable solutions to your questions quickly and accurately with help from our dedicated community of experts.

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