Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 1 - Functions - 1.1 Review of Functions - 1.1 Exercises: 72

Answer

This function is only symmetric about the origin.
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Work Step by Step

$f(x)=3x^{5}+2x^{3}-x$ $\textbf{Symmetry about the $x$-axis}$ This function has no symmetry about the x-axis, because if it had, it would violate the Vertical Rule Test. Another way to realize this is that changing $f(x)$ by $−f(x)$ does not yield an equivalent function. $\textbf{Symmetry about the $y$-axis}$ Substitute $x$ by $-x$ in $f(x)$ and simplify: $f(-x)=3(-x)^{5}+2(-x)^{3}-(-x)=...$ $...=-3x^{5}-2x^{3}+x$ Since $f(-x)\ne f(x)$, the given function is not symmetric about the $y$-axis. $\textbf{Symmetry about the origin}$ It can be seen in the test for $y$-axis symmetry that $f(-x)=-f(x)$. Because of this, this function is symmetric about the origin. The graph of this function is:
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