Algebra Examples

$f (x) = \frac{x^{3} - 2 x}{x^{2} - 1}$

Step 1

Determine if the function is odd, even, or neither in order to find the symmetry.

1. If odd, the function is symmetric about the origin.

2. If even, the function is symmetric about the y-axis.

Step 2

Simplify.

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Step 2.1

Factor $x$ out of $x^{3} - 2 x$ .

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Step 2.1.1

Factor $x$ out of $x^{3}$ .

$f (x) = \frac{x \cdot x^{2} - 2 x}{x^{2} - 1}$

Step 2.1.2

Factor $x$ out of $- 2 x$ .

$f (x) = \frac{x \cdot x^{2} + x \cdot - 2}{x^{2} - 1}$

Step 2.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 2$ .

$f (x) = \frac{x (x^{2} - 2)}{x^{2} - 1}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

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

$f (x) = \frac{x (x^{2} - 2)}{x^{2} - 1^{2}}$

Step 2.2.2

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 $b = 1$ .

$f (x) = \frac{x (x^{2} - 2)}{(x + 1) (x - 1)}$

Step 3

Find $f (- x)$ .

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Step 3.1

Find $f (- x)$ by substituting $- x$ for all occurrence of $x$ in $f (x)$ .

$f (- x) = \frac{(- x) ({(- x)}^{2} - 2)}{((- x) + 1) ((- x) - 1)}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Apply the product rule to $- x$ .

$f (- x) = \frac{- x ({(- 1)}^{2} x^{2} - 2)}{(- x + 1) (- x - 1)}$

Step 3.2.2

Raise $- 1$ to the power of $2$ .

$f (- x) = \frac{- x (1 x^{2} - 2)}{(- x + 1) (- x - 1)}$

Step 3.2.3

Multiply $x^{2}$ by $1$ .

$f (- x) = \frac{- x (x^{2} - 2)}{(- x + 1) (- x - 1)}$

Step 3.3

Simplify with factoring out.

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Step 3.3.1

Move the negative in front of the fraction.

$f (- x) = - \frac{x (x^{2} - 2)}{(- x + 1) (- x - 1)}$

Step 3.3.2

Factor $- 1$ out of $- x$ .

$f (- x) = - \frac{x (x^{2} - 2)}{(- (x) + 1) (- x - 1)}$

Step 3.3.3

Rewrite $1$ as $- 1 (- 1)$ .

$f (- x) = - \frac{x (x^{2} - 2)}{(- (x) - 1 \cdot - 1) (- x - 1)}$

Step 3.3.4

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

$f (- x) = - \frac{x (x^{2} - 2)}{- (x - 1) (- x - 1)}$

Step 3.3.5

Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

$f (- x) = - \frac{x (x^{2} - 2)}{- 1 (x - 1) (- x - 1)}$

Step 3.3.6

Factor $- 1$ out of $- x$ .

$f (- x) = - \frac{x (x^{2} - 2)}{- 1 (x - 1) (- (x) - 1)}$

Step 3.3.7

Rewrite $- 1$ as $- 1 (1)$ .

$f (- x) = - \frac{x (x^{2} - 2)}{- 1 (x - 1) (- (x) - 1 \cdot 1)}$

Step 3.3.8

Factor $- 1$ out of $- (x) - 1 (1)$ .

$f (- x) = - \frac{x (x^{2} - 2)}{- 1 (x - 1) (- (x + 1))}$

Step 3.3.9

Simplify the expression.

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Step 3.3.9.1

Rewrite $- (x + 1)$ as $- 1 (x + 1)$ .

$f (- x) = - \frac{x (x^{2} - 2)}{- 1 (x - 1) (- 1 (x + 1))}$

Step 3.3.9.2

Multiply $- 1$ by $- 1$ .

$f (- x) = - \frac{x (x^{2} - 2)}{1 (x - 1) (x + 1)}$

Step 3.3.9.3

Multiply $x - 1$ by $1$ .

$f (- x) = - \frac{x (x^{2} - 2)}{(x - 1) (x + 1)}$

Step 4

A function is even if $f (- x) = f (x)$ .

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Step 4.1

Check if $f (- x) = f (x)$ .

Step 4.2

Since $- \frac{x (x^{2} - 2)}{(x - 1) (x + 1)}$ $\neq$ $\frac{x (x^{2} - 2)}{(x + 1) (x - 1)}$ , the function is not even.

The function is not even

Step 5

A function is odd if $f (- x) = - f (x)$ .

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Step 5.1

Multiply $- 1$ by $\frac{x (x^{2} - 2)}{(x + 1) (x - 1)}$ .

$- f (x) = - \frac{x (x^{2} - 2)}{(x + 1) (x - 1)}$

Step 5.2

Since $- \frac{x (x^{2} - 2)}{(x - 1) (x + 1)} = - \frac{x (x^{2} - 2)}{(x + 1) (x - 1)}$ , the function is odd.

The function is odd

Step 6

Since the function is odd, it is symmetric about the origin.

Origin Symmetry

Step 7

Since the function is not even, it is not symmetric about the y-axis.

No y-axis symmetry

Step 8

Determine the symmetry of the function.

Origin symmetry

Step 9