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.
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.
