Chapter 1, Problem 13RE

In Problems 10-14, test each equation for symmetry with respect to the $x-axis$, the $y-axis$, and the origin.

$y=x^3-x$

To determine

To test: The equation $y=x³-x$ for symmetry, with respect to the $x\text{-axis}$, $y\text{-axis}$ and the origin.

Answer to Problem 13RE

Solution:

$\text{The graph of the given equation is not symmetric, with respect to }x\text{-axis, }y\text{-axis and the origin}$.

Given:

The equation $y=x³-x$

Explanation of Solution

Replace $y$ by $-y$ in the equation $y=x³-x$.

$\left(-y\right)=x³-x$

$-y=x³-x$, which is not equivalent to the given equation.

Hence the graph is not symmetric, with respect to the $x\text{-axis}$.

Replace $x$ by $-x$.

$y=\left(-x\right)³-\left(-x\right)$

$y=x³+x$, which is not equivalent to the given equation.

Hence the graph is not symmetric, with respect to the $y\text{-axis}$.

Replace $x$ by $-x$ and $y$ by $-y$.

$\left(-y\right)=\left(-x\right)³-\left(-x\right)$

$-y=-x³+x$, which is not equivalent to the given equation.

Hence the graph is not symmetric, with respect to the origin.