Find the Symmetry (x^2-1)/x

$\frac{x^{2} - 1}{x}$

Step 1

Write $\frac{x^{2} - 1}{x}$ as an equation.

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

Step 2

There are three types of symmetry:

1. X-Axis Symmetry

2. Y-Axis Symmetry

3. Origin Symmetry

Step 3

If $(x, y)$ exists on the graph, then the graph is symmetric about the:

1. X-Axis if $(x, - y)$ exists on the graph

2. Y-Axis if $(- x, y)$ exists on the graph

3. Origin if $(- x, - y)$ exists on the graph

Step 4

Simplify the numerator.

Rewrite $1$ as $1^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$y = \frac{(x + 1) (x - 1)}{x}$

Step 5

Check if the graph is symmetric about the $x$ -axis by plugging in $- y$ for $y$ .

$- y = \frac{(x + 1) (x - 1)}{x}$

Step 6

Since the equation is not identical to the original equation, it is not symmetric to the x-axis.

Not symmetric to the x-axis

Step 7

Check if the graph is symmetric about the $y$ -axis by plugging in $- x$ for $x$ .

$y = \frac{(- x + 1) (- x - 1)}{- x}$

Step 8

Move the negative in front of the fraction.

$y = - \frac{(- x + 1) (- x - 1)}{x}$

Step 9

Since the equation is not identical to the original equation, it is not symmetric to the y-axis.

Not symmetric to the y-axis

Step 10

Check if the graph is symmetric about the origin by plugging in $- x$ for $x$ and $- y$ for $y$ .

$- y = \frac{(- x + 1) (- x - 1)}{- x}$

Step 11

Move the negative in front of the fraction.

$- y = - \frac{(- x + 1) (- x - 1)}{x}$

Step 12

Multiply both sides by $- 1$ .

Multiply each term by $- 1$ .

$- - y = - - \frac{(- x + 1) (- x - 1)}{x}$

Multiply $- - y$ .

Multiply $- 1$ by $- 1$ .

$1 y = - - \frac{(- x + 1) (- x - 1)}{x}$

Multiply $y$ by $1$ .

$y = - - \frac{(- x + 1) (- x - 1)}{x}$

Multiply $- - \frac{(- x + 1) (- x - 1)}{x}$ .

Multiply $- 1$ by $- 1$ .

$y = 1 \frac{(- x + 1) (- x - 1)}{x}$

Multiply $\frac{(- x + 1) (- x - 1)}{x}$ by $1$ .

$y = \frac{(- x + 1) (- x - 1)}{x}$

Step 13

Since the equation is identical to the original equation, it is symmetric to the origin.

Symmetric with respect to the origin

Step 14

Find the Symmetry (x^2-1)/x

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