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Determine whether the function
f(x)=1-x^(4)+6x^(6) is even, odd or neither.
odd
even
neither

Determine whether the function $f(x)=1-x^{4}+6 x^{6}$ is even, odd or neither. $\newline$ odd $\newline$ even $\newline$ neither

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Even and odd functions

Full solution

Q. Determine whether the function

f(x)=1-x^{4}+6 x^{6}

is even, odd or neither.

\newline

odd

\newline

even

\newline

neither

Identify Function Type: To determine if the function is even, odd, or neither, we need to compare $f(x)$ with $f(-x)$ . If $f(-x) = f(x)$ , the function is even. If $f(-x) = -f(x)$ , the function is odd. If neither condition is met, the function is neither even nor odd.
Substitute $-x$ : First, let's find $f(-x)$ by substituting $-x$ for $x$ in the function $f(x)=1-x^{4}+6x^{6}$ . $\newline$ $f(-x) = 1-(-x)^{4}+6(-x)^{6}$
Simplify $f(-x)$ : Now, simplify the expression for $f(-x)$ .
$f(-x) = 1-((-x)^{4})+6((-x)^{6})$
$f(-x) = 1-(x^4)+6(x^6)$
$f(-x) = 1-x^4+6x^6$
Compare $f(x)$ and $f(-x)$ : We have the original function $f(x)=1-x^{4}+6x^{6}$ and the transformed function $f(-x)=1-(-x)^{4}+6(-x)^{6}$ . Comparing $f(x)$ and $f(-x)$ , we see that they are identical. $\newline$ $f(x) = f(-x)$
Conclusion: Since $f(-x) = f(x)$ , the function $f(x)$ is an even function.

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