Precalculus Examples

$f (x) = \frac{- x^{2}}{x^{4} - 25}$

Step 1

Simplify.

Tap for more steps...

Step 1.1

Simplify the denominator.

Tap for more steps...

Step 1.1.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$f (x) = \frac{- x^{2}}{{(x^{2})}^{2} - 25}$

Step 1.1.2

Rewrite $25$ as $5^{2}$ .

$f (x) = \frac{- x^{2}}{{(x^{2})}^{2} - 5^{2}}$

Step 1.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 5$ .

$f (x) = \frac{- x^{2}}{(x^{2} + 5) (x^{2} - 5)}$

Step 1.2

Move the negative in front of the fraction.

$f (x) = - \frac{x^{2}}{(x^{2} + 5) (x^{2} - 5)}$

Step 2

Find $f (- x)$ .

Tap for more steps...

Step 2.1

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

$f (- x) = - \frac{{(- x)}^{2}}{({(- x)}^{2} + 5) ({(- x)}^{2} - 5)}$

Step 2.2

Simplify the numerator.

Tap for more steps...

Step 2.2.1

Apply the product rule to $- x$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

$f (- x) = - \frac{x^{2}}{({(- x)}^{2} + 5) ({(- x)}^{2} - 5)}$

Step 2.3

Simplify the denominator.

Tap for more steps...

Step 2.3.1

Apply the product rule to $- x$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

$f (- x) = - \frac{x^{2}}{(x^{2} + 5) ({(- x)}^{2} - 5)}$

Step 2.3.4

Apply the product rule to $- x$ .

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

Step 2.3.5

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

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

Step 2.3.6

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

$f (- x) = - \frac{x^{2}}{(x^{2} + 5) (x^{2} - 5)}$

Step 3

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

Tap for more steps...

Step 3.1

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

Step 3.2

Since $- \frac{x^{2}}{(x^{2} + 5) (x^{2} - 5)} = - \frac{x^{2}}{(x^{2} + 5) (x^{2} - 5)}$ , the function is even.

The function is even

Step 4