Precalculus (6th Edition) Blitzer

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

Chapter 1 - Section 1.3 - More on Functions and Their Graphs - Exercise Set - Page 196: 44

Answer

The function is an even function and the graph of the function is symmetric about the $y-\text{axis}$.

Work Step by Step

An odd function is the function having symmetry with the origin and an even function is the function having symmetry with the y-axis. To check whether the given function is even or odd, substitute x with $-x$ in the provided function and simplify as follows: $\begin{align} & f\left( -x \right)=2{{\left( -x \right)}^{2}}+{{\left( -x \right)}^{4}}+1 \\ & f\left( -x \right)=2{{x}^{2}}+{{x}^{4}}+1 \\ & f\left( -x \right)=f\left( x \right) \end{align}$ On putting –x in the place of x, it can be seen that the same function is obtained. The graph of the function is symmetric about the $y-\text{axis}$ and thus the definition of even function is fulfilled. Hence, the function is an even function and the graph of the function is symmetric about the $y-\text{axis}$.
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