Chapter 1.2, Problem 38E

To determine

ToCheck:The symmetry with respect to both axes and the origin using the algebraic tests for the equation $y=\frac{1}{x^2+1}$

Answer to Problem 38E

The equation $y=\frac{1}{x^2+1}$ is not symmetric with respect to $x-\text{axis}$ .

The equation $y=\frac{1}{x^2+1}$ is symmetric with respect to $y-\text{axis}$ .

The equation $y=\frac{1}{x^2+1}$ is not symmetric with respect to origin.

Explanation of Solution

Given:

The equation $y=\frac{1}{x^2+1}$

Concept Used:

The graph of an equation is symmetric with respect to $x-\text{axis}$ when replacing $y$ with $-y$ yields an equivalent equation.

The graph of an equation is symmetric with respect to $y-\text{axis}$ when replacing $x$ with $-x$ yields an equivalent equation.

The graph of an equation is symmetric with respect to origin when replacing $x$ with $-x$ and $y$ with $-y$ yields an equivalent equation.

Calculation:

Given the equation $y=\frac{1}{x^2+1}$

Checking with respect to $x-\text{axis}$

Replacing $y$ with $-y$ in the given equation

that is $\begin{array}{c}\left(-y\right)=\frac{1}{x^2+1}\text{ Replacing }y\text{ with }-y\\ -y=\frac{1}{x^2+1}\text{ Not an Equivalent equation}\end{array}$

The equation $y=\frac{1}{x^2+1}$ is not symmetric with respect to $x-\text{axis}$ .

Checking with respect to $y-\text{axis}$

Replacing $x$ with $-x$ in the given equation

that is $\begin{array}{c}y=\frac{1}{{\left(-x\right)}^2+1}\text{ Replacing }x\text{ with }-x\\ y=\frac{1}{x^2+1}\text{ Equivalent equation}\end{array}$

The equation $y=\frac{1}{x^2+1}$ is symmetric with respect to $y-\text{axis}$ .

Checking with respect to origin

Replacing $x$ with $-x$ and $y$ with $-y$ in the given equation

that is $\begin{array}{c}\left(-y\right)=\frac{1}{{\left(-x\right)}^2+1}\text{ Replacing }x\text{ with }-x\text{ and }y\text{ with }-y\\ -y=\frac{1}{x^2+1}\text{ Not an equivalent equation}\end{array}$

The equation $y=\frac{1}{x^2+1}$ is notsymmetric with respect to origin.

Therefore, theequation $y=\frac{1}{x^2+1}$ is symmetric with respect to $y-\text{axis}$ only.

Conclusion:

The equation $y=\frac{1}{x^2+1}$ is not symmetric with respect to $x-\text{axis}$ .

The equation $y=\frac{1}{x^2+1}$ is symmetric with respect to $y-\text{axis}$ .

The equation $y=\frac{1}{x^2+1}$ is not symmetric with respect to origin.