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Chapter 1: Problem 38

Use the algebraic tests to check for symmetry with respect to both axes and the origin. $$y=\frac{1}{x^{2}+1}$$

Short Answer

Expert verified
The function \(y=\frac{1}{x^{2}+1}\) is symmetric about the y-axis only, not about the x-axis or the origin.

Step by step solution

01

Testing Symmetry about the Y-axis

Replace every 'x' in the equation with '-x', then simplify the expression. If the simplified equation is equivalent to the original function, the function is symmetric about the y-axis. Here, replacing 'x' in the original function \(y=\frac{1}{x^{2}+1}\) with '-x' gives \(y=\frac{1}{(-x)^{2}+1}\), which simplifies back to \(y=\frac{1}{x^{2}+1}\), the original function.
02

Testing Symmetry about the X-axis

To test for x-axis symmetry, replace 'y' with '-y' in the function. If the simplified equation is equivalent to the original function, then it's symmetric about the x-axis. However, in our case, replacing 'y' with '-y' gives \(-y=\frac{1}{x^{2}+1}\). This is not equivalent to the original function, thus confirming that the function is not symmetric about the x-axis.
03

Testing Symmetry about the Origin

To check for Origin symmetry, replace both 'x' with '-x' and 'y' with '-y'. If the resulting equation simplifies to the original function, then the function is symmetric about the origin. But when we replace 'y' with '-y' and 'x' with '-x' in \(y=\frac{1}{x^{2}+1}\), we get \(-y=\frac{1}{(-x)^{2}+1}\), which does not simplify back to the original function, hence the function is not symmetric about the origin.
04

Summary

So, after testing, it is seen that the function is only symmetric about the y-axis (as it remains invariant under 'x' to '-x' transformation), but not about the x-axis or the origin.

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