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Determine whether the function
f(x)=x^(4)+5x^(6)-6 is even, odd or neither.
even
neither
odd

Determine whether the function $f(x)=x^{4}+5 x^{6}-6$ is even, odd or neither. $\newline$ even $\newline$ neither $\newline$ odd

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Even and odd functions

Full solution

Q. Determine whether the function

f(x)=x^{4}+5 x^{6}-6

is even, odd or neither.

\newline

even

\newline

neither

\newline

odd

Consider function $f(x)$ : Consider the function $f(x) = x^4 + 5x^6 - 6$ . To determine if the function is even, odd, or neither, we need to evaluate $f(-x)$ and compare it to $f(x)$ . Substitute $-x$ for $x$ in $f(x)$ to get $f(-x)$ . $f(-x) = (-x)^4 + 5(-x)^6 - 6$
Simplify $f(-x)$ : Simplify the right side of the function $f(-x)$ .
$f(-x) = (-x)^4 + 5(-x)^6 - 6$
Since both $x^4$ and $x^6$ are even powers, $(-x)^4 = x^4$ and $(-x)^6 = x^6$ .
Therefore, $f(-x)$ simplifies to:
$f(-x) = x^4 + 5x^6 - 6$
Compare $f(-x)$ with $f(x)$ : Compare $f(-x)$ with $f(x)$ . We have $f(x) = x^4 + 5x^6 - 6$ and $f(-x) = x^4 + 5x^6 - 6$ . Since $f(-x) = f(x)$ , the function $f(x)$ is an even function.

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