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 - Page 11: 71


The function is only symmetric about the $y$-axis

Work Step by Step

$f(x)=x^{4}+5x^{2}-12$ $\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: $f(-x)=(-x)^{4}+5(-x)^{2}-12=...$ $...=x^{4}+5x^{2}-12$ Since $f(-x)=f(x)$, this function is symmetric about the $y$-axis. $\textbf{Symmetry about the origin}$ This function has no symmetry about the origin beacuse, as it was seen when testing for $y$-axis symmetry, $f(-x)$ is not equal to $-f(x)$. The graph of this function is:
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