Calculus Examples

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

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

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 the denominator.

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Step 2.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

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

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

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 1

$b = 1$ .

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

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

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

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

Step 3

Find

f (- x)

$f (- x)$ .

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Step 3.1

Find

f (- x)

$f (- x)$ by substituting

- x

$- x$ for all occurrence of

x

$x$ in

f (x)

$f (x)$ .

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

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

Apply the product rule to

- x

$- x$ .

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

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

Step 3.2.2

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

Step 3.2.3

Multiply

x^{2}

$x^{2}$ by

1

$1$ .

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

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

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

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

Step 3.3

Simplify with factoring out.

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Step 3.3.1

Factor

- 1

$- 1$ out of

- x

$- x$ .

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

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

Step 3.3.2

Rewrite

1

$1$ as

- 1 (- 1)

$- 1 (- 1)$ .

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

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

Step 3.3.3

Factor

- 1

$- 1$ out of

- (x) - 1 (- 1)

$- (x) - 1 (- 1)$ .

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

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

Step 3.3.4

Rewrite

- (x - 1)

$- (x - 1)$ as

- 1 (x - 1)

$- 1 (x - 1)$ .

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

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

Step 3.3.5

Factor

- 1

$- 1$ out of

- x

$- x$ .

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

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

Step 3.3.6

Rewrite

- 1

$- 1$ as

- 1 (1)

$- 1 (1)$ .

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

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

Step 3.3.7

Factor

- 1

$- 1$ out of

- (x) - 1 (1)

$- (x) - 1 (1)$ .

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

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

Step 3.3.8

Simplify the expression.

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Step 3.3.8.1

Rewrite

- (x + 1)

$- (x + 1)$ as

- 1 (x + 1)

$- 1 (x + 1)$ .

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

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

Step 3.3.8.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 3.3.8.3

Multiply

x - 1

$x - 1$ by

1

$1$ .

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

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

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

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

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

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

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

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

Step 4

A function is even if

f (- x) = f (x)

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

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Step 4.1

Check if

f (- x) = f (x)

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

Step 4.2

Since

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

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

\neq

$\neq$

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

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

The function is not even

Step 5

A function is odd if

f (- x) = - f (x)

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

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Step 5.1

Multiply

- 1

$- 1$ by

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

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

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

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

Step 5.2

Since

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

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

\neq

$\neq$

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

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

The function is not odd

Step 6

The function is neither odd nor even

Step 7

Since the function is not odd, it is not symmetric about the origin.

No origin symmetry

Step 8

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

No y-axis symmetry

Step 9

Since the function is neither odd nor even, there is no origin / y-axis symmetry.

Function is not symmetric

Step 10