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Determine whether the function
f(x)=8x^(7)-3x^(5)+3x is even, odd or neither.
even
neither
odd

Determine whether the function $f(x)=8 x^{7}-3 x^{5}+3 x$ 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)=8 x^{7}-3 x^{5}+3 x

is even, odd or neither.

\newline

even

\newline

neither

\newline

odd

Define function $f(x)$ : Define the function $f(x)$ . The given function is $f(x) = 8x^{7} - 3x^{5} + 3x$ . To determine if the function is even, odd, or neither, we need to compare $f(x)$ with $f(-x)$ .
Calculate $f(-x)$ : Calculate $f(-x)$ . $\newline$ Substitute $-x$ for $x$ in $f(x)=8x^{7}-3x^{5}+3x$ to get $f(-x)$ . $\newline$ $f(-x)=8(-x)^{7}-3(-x)^{5}+3(-x)$ $\newline$ Now simplify the right side of the function. $\newline$ $f(-x)=8(-1)^7\cdot x^{7}-3(-1)^5\cdot x^{5}+3(-1)\cdot x$ $\newline$ $f(-x)=-8x^{7}+3x^{5}-3x$
Compare $f(x)$ with $f(-x)$ : Compare $f(x)$ with $f(-x)$ . We have: $f(x)=8x^{7}-3x^{5}+3x$ $f(-x)=-8x^{7}+3x^{5}-3x$ Since $f(-x)$ is not equal to $f(x)$ and $f(-x)$ is not equal to $-f(x)$ , the function $f(x)$ is neither even nor odd.

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