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Question

# Determine whether the graph of each function is symmetric with respect to the origin. $f(x)=\frac{1}{4 x^{7}}$

Solution

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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)&=\frac{1}{4x^7}\to -f(x)=\frac{1}{4\left(-x\right)^7} \to -f(x)=-\frac{1}{4x^7}\\ f(x)&=\frac{1}{4x^7}\\ \end{align*}

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

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