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Question

# Test for symmetry with respect to each axis and to the origin. $y=\frac{x}{x^{2}+1}$

Solution

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1 of 2

We have $y= \dfrac{x}{x^2+1}$

$\color{#4257b2}\text{First: we'll test the equation for symmetry with respect to the x-axis}$:

$(-y) = \dfrac{x}{x^2+1}$( Replace y by -y)

$\Rightarrow y=-\dfrac{x}{x^2+1}$

The result is not an equivalent equation.

Therefore the graph is not symmetric with respect to the x-axis

$\color{#4257b2}\text{Second: we'll test the equation for symmetry with respect to the y-axis}$:

$y =\dfrac{-x}{(-x)^2+1}$( Replace x by -x)

$\Rightarrow y=-\dfrac{x}{x^2+1}$

The result is not an equivalent equation.

Therefore the graph is not symmetric with respect to the y-axis

$\color{#4257b2}\text{Third: we'll test the equation for symmetry with respect to the origin}$:

$(-y) = \dfrac{-x}{(-x)^2+1}$( Replace y by -y and x by -x)

$\Rightarrow -y=-\dfrac{x}{x^2+1}$

The result is an equivalent equation.

Therefore the graph is symmetric with respect to the origin

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