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Determine whether the function
f(x)=3x-7x^(5)+2x^(2) is even, odd or neither.
odd
neither
even

Determine whether the function $f(x)=3 x-7 x^{5}+2 x^{2}$ is even, odd or neither. $\newline$ odd $\newline$ neither $\newline$ even

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Even and odd functions

Full solution

Q. Determine whether the function

f(x)=3 x-7 x^{5}+2 x^{2}

is even, odd or neither.

\newline

odd

\newline

neither

\newline

even

Write Original Function: To determine if the function is even, odd, or neither, we need to evaluate $f(-x)$ and compare it to $f(x)$ . $\newline$ First, let's write down the original function: $\newline$ $f(x) = 3x - 7x^5 + 2x^2$ $\newline$ Now, let's substitute $-x$ for $x$ in the function to find $f(-x)$ . $\newline$ $f(-x) = 3(-x) - 7(-x)^5 + 2(-x)^2$
Substitute $-x$ for $x$ : Next, we simplify the expression for $f(-x)$ .
$f(-x) = -3x - 7(-x)^5 + 2x^2$
Since the exponent $5$ is odd, $(-x)^5 = -(x^5)$ . The exponent $2$ is even, so $(-x)^2 = x^2$ .
$f(-x) = -3x + 7x^5 + 2x^2$
Simplify $f(-x)$ Expression: Now we compare $f(-x)$ with $f(x)$ .
$f(x) = 3x - 7x^5 + 2x^2$
$f(-x) = -3x + 7x^5 + 2x^2$
We can see that $f(-x)$ is not equal to $f(x)$ and also not equal to $-f(x)$ , because the signs of the terms with $x$ and $x^5$ are different.
Therefore, the function is neither even nor odd.

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