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

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

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Even and odd functions

Full solution

Q. Determine whether the function

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

is even, odd or neither.

\newline

even

\newline

odd

\newline

neither

Determine 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. $\newline$ Let's start by finding $f(-x)$ . $\newline$ $f(x) = -6x^6 + 1 + x^4$ $\newline$ $f(-x) = -6(-x)^6 + 1 + (-x)^4$
Find $f(-x)$ : Now we simplify $f(-x)$ .
$f(-x) = -6(-x)^6 + 1 + (-x)^4$
Since both $x^6$ and $x^4$ are even powers, $(-x)^6 = x^6$ and $(-x)^4 = x^4$ .
$f(-x) = -6x^6 + 1 + x^4$
Simplify $f(-x)$ : We compare $f(-x)$ with $f(x)$ .
$f(x) = -6x^6 + 1 + x^4$
$f(-x) = -6x^6 + 1 + x^4$
Since $f(-x) = f(x)$ , the function $f(x)$ is even.

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