Chapter 1.2, Problem 70AYU

In Problems 49-64, list the intercepts and test for symmetry.

$y=\frac{x^4+1}{2x^5}$

To determine

To find: The intercepts and test for the symmetry.

Answer to Problem 70AYU

The $y\text{-intercept}$ is not defined. The graph of the equation is symmetric with respect to origin..

Explanation of Solution

Given:

The equation $y=\frac{x^4+1}{2x^5}$

To find the $x\text{-intercept(s)}$, let $y=0$

Therefore   $0=\frac{x^4+1}{2x^5}$

This implies $x^4+1=0$

$x^4+1-1=0-1$

$x^4=-1$

$x=\sqrt[4]{-1}$   not real

To find the $y\text{-intercept(s)}$, let $x=0$

Therefore   $y=\frac{{(0)}^4+1}{2{(0)}^5}$

$y=\frac{1}{0}$

$y\text{ = not defined}$

The $y\text{-intercept}$ is not defined.

Now test the equation for symmetry with respect to $x\text{-axis}$. Replace $y$ by $-y$.

$(-y)=\frac{x^4+1}{2x^5}$ is not the same as the original equation when $y$ is replaced by $-y$.

Therefore the graph of the equation is not symmetric with respect to $x\text{-axis}$.

Now test the equation for symmetry with respect to $y\text{-axis}$. Replace $x$ by $-x$.

$y=\frac{{(-x)}^4+1}{2{(-x)}^5}$

$y=\frac{x^4+1}{-2x^5}$ is not equivalent to the original equation.

Therefore the graph of the equation is not symmetric with respect to $y\text{-axis}$.

Now test the equation for symmetry with respect to the origin. Replace $x$ by $-x$ and $y$ by $-y$.

$\left(-y\right)=\frac{{(-x)}^4+1}{2{(-x)}^5}$

$-y=\frac{x^4+1}{-2x^5}$

$y=\frac{x^4+1}{2x^5}$is equivalent to the original equation. Therefore the graph of the equation is symmetric with respect to the origin.