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Let [tex]f(x) = 7 - 7x^4 + x^6[/tex]. Determine whether the function is odd, even, or neither.

Answer: [tex]\square[/tex]

Answer :

To determine whether the function [tex]\( f(x) = 7 - 7x^4 + x^6 \)[/tex] is odd, even, or neither, follow these steps:

1. Definitions:
- A function is even if for all [tex]\( x \)[/tex], [tex]\( f(x) = f(-x) \)[/tex].
- A function is odd if for all [tex]\( x \)[/tex], [tex]\( f(-x) = -f(x) \)[/tex].

2. Find [tex]\( f(-x) \)[/tex]:
Substitute [tex]\(-x\)[/tex] into the function:
[tex]\[
f(-x) = 7 - 7(-x)^4 + (-x)^6
\][/tex]

3. Simplify [tex]\( f(-x) \)[/tex]:
- Calculate [tex]\((-x)^4 = x^4\)[/tex] because any even power negates the negative sign.
- Calculate [tex]\((-x)^6 = x^6\)[/tex] for the same reason.
- Substitute these back in:
[tex]\[
f(-x) = 7 - 7x^4 + x^6
\][/tex]

4. Compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
- We have [tex]\( f(x) = 7 - 7x^4 + x^6 \)[/tex] and [tex]\( f(-x) = 7 - 7x^4 + x^6 \)[/tex].
- Since [tex]\( f(x) = f(-x) \)[/tex], the function is even.

Thus, the function [tex]\( f(x) = 7 - 7x^4 + x^6 \)[/tex] is an even function.

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