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Question

# Determine whether the graph of each function is symmetric with respect to the origin. $f(x)=-7 x^{5}+8 x$

Solution

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Step 1
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To check for $\textbf{origin symmetry}$, we need to replace $x$ with $-x$ and $f(x)$ with $-f(x)$. If we get an equal expression, the function is $\textbf{symmetrical}$ in regards to the $\textbf{origin}$.

\begin{align*} f(x)&=-7x^5 + 8x \to -f(x)=-7\left(-x\right)^5 -8x \to -f(x)&=7x^5-8x\\ f(x)&=-7x^5+8x \end{align*}

Since we got a valid expression, $f(x)$ is $\textbf{symmetrical in regards to the origin}$.

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