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Chapter 1: Problem 71

Determine whether the function is even, odd, or neither. Then describe the symmetry. $$f(x)=x^{6}-2 x^{2}+3$$

Short Answer

Expert verified
The given function \(f(x) = x^{6} - 2x^{2} + 3\) is an even function. This means that it is symmetric about the y-axis.

Step by step solution

01

Write down the function

The given function is \(f(x) = x^{6} - 2x^{2} + 3\).
02

Substitute x by -x

Now, let's substitute \(x\) by \(-x\) in the function to get \(f(-x)\). This yields: \(f(-x) = (-x)^{6} - 2(-x)^{2} + 3\), which simplifies to \(f(-x) = x^{6} - 2x^{2} + 3\).
03

Compare f(x) and f(-x)

Comparing the original function \(f(x)\) and the function \(-f(x)\) after the substitution, we see that \(f(x) = f(-x)\). Therefore, the function is even.
04

Describe the symmetry

Since the function is even, it means that it is symmetric about the y-axis.

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