Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 1 - Review Exercises - Page 301: 43

Answer

The given function is an even function and has symmetry about the y-axis only.

Work Step by Step

Step I: To check if even or odd. In the given equation, if $f\left( -x \right)=f\left( x \right)$ , then it is an even function, and if $f\left( -x \right)=-f\left( x \right)$ , then it is odd. $\begin{align} & f\left( x \right)={{x}^{4}}-2{{x}^{2}}+1 \\ & f\left( -x \right)={{\left( -x \right)}^{4}}-2{{\left( -x \right)}^{2}}+1 \\ & ={{x}^{4}}-2{{x}^{2}}+1 \\ & =f\left( x \right). \end{align}$ It is an even function. Step II: To check symmetry about the y-axis: Putting $x=-x$ in the given equation, if the equation remains the same, then it has symmetry about the y-axis. $\begin{align} & f\left( x \right)={{x}^{4}}-2{{x}^{2}}+1 \\ & f\left( -x \right)={{\left( -x \right)}^{4}}-2{{\left( -x \right)}^{2}}+1 \\ & ={{x}^{4}}-2{{x}^{2}}+1 \end{align}$ It is the same as the provided equation above, hence it has symmetry about the y-axis. Step III: To check symmetry about the x-axis: Putting $y=-y$ in the given equation, if the equation remains the same, then it has symmetry about the x-axis. $\begin{align} & y={{x}^{4}}-2{{x}^{2}}+1 \\ & \left( -y \right)={{x}^{4}}-2{{x}^{2}}+1 \\ & -y={{x}^{4}}-2{{x}^{2}}+1 \\ & y=-\left( {{x}^{4}}-2{{x}^{2}}+1 \right) \end{align}$ It is not the same as the provided equation above, hence it is not symmetric about the x-axis. Step IV: To check symmetry about the origin: Putting $x=-x\text{ and }y=-y$ in the given equation, if the equation remains the same, then it has symmetry about the origin. $\begin{align} & y={{x}^{4}}-2{{x}^{2}}+1 \\ & \left( -y \right)={{\left( -x \right)}^{4}}-2{{\left( -x \right)}^{2}}+1 \\ & -y={{x}^{4}}-2{{x}^{2}}+1 \\ & y=-\left( {{x}^{4}}-2{{x}^{2}}+1 \right) \end{align}$ It is not the same as the provided equation above, hence it has no symmetry about the origin. Therefore, the given function is an even function and has symmetry about the y-axis only.
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