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Chapter 1: Problem 37

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

Short Answer

Expert verified
The function \(y=\frac{x}{x^{2}+1}\) does not have symmetry about the y-axis, it can't be tested for symmetry about the x-axis, but it has symmetry about the origin.

Step by step solution

01

Test for symmetry about the y-axis

For a function to be symmetric about the y-axis, we have to replace x by -x in the function equation and check if the function remains the same i.e., we aim to show that f(-x) = f(x). Let us substitute x with -x in the function \(y=\frac{x}{x^{2}+1}\), we get \(y=\frac{-x}{(-x)^{2}+1}\) which simplifies to \(y=\frac{-x}{x^{2}+1}\). Comparing with the original function, we can see that \(f(-x) \neq f(x)\). Therefore, the function is not symmetric about the y-axis.
02

Test for symmetry about the x-axis

For a function to be symmetric about the x-axis, we have to replace y by -y in the function equation and check if the expression remains the same i.e., we aim to show that f(-y) = f(y). However, the function \(y=\frac{x}{x^{2}+1}\) is not explicitly solved for x, so it is not possible to perform a test for x-axis symmetry.
03

Test for symmetry about the origin

For a function to be symmetric about the origin, we have to replace x by -x in the function equation and check if we can obtain the negation of the original function i.e., we aim to show that f(-x) = -f(x). We return to our y value when we replaced x by -x: \(y=\frac{-x}{x^{2}+1}\). Multiplying the original function by -1 we get \(-y=\frac{-x}{x^{2}+1}\). Since we find that f(-x) = -f(x), we can conclude that the function is symmetric about the origin.

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