Precalculus Examples

$- x^{4} + x$

Step 1

Write $- x^{4} + x$ as a function.

$f (x) = - x^{4} + x$

Step 2

Find $f (- x)$ .

Tap for more steps...

Step 2.1

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

$f (- x) = - {(- x)}^{4} - x$

Step 2.2

Remove parentheses.

$f (- x) = - {(- x)}^{4} - x$

Step 2.3

Simplify each term.

Tap for more steps...

Step 2.3.1

Apply the product rule to $- x$ .

$f (- x) = - ({(- 1)}^{4} x^{4}) - x$

Step 2.3.2

Multiply $- 1$ by ${(- 1)}^{4}$ by adding the exponents.

Tap for more steps...

Step 2.3.2.1

Move ${(- 1)}^{4}$ .

$f (- x) = {(- 1)}^{4} \cdot (- 1 x^{4}) - x$

Step 2.3.2.2

Multiply ${(- 1)}^{4}$ by $- 1$ .

Tap for more steps...

Step 2.3.2.2.1

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

$f (- x) = {(- 1)}^{4} \cdot ((- 1) x^{4}) - x$

Step 2.3.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- x) = {(- 1)}^{4 + 1} x^{4} - x$

Step 2.3.2.3

Add $4$ and $1$ .

$f (- x) = {(- 1)}^{5} x^{4} - x$

Step 2.3.3

Raise $- 1$ to the power of $5$ .

$f (- x) = - x^{4} - x$

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 $- x^{4} - x$ $\neq$ $- x^{4} + x$ , the function is not even.

The function is not even

Step 4

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

Tap for more steps...

Step 4.1

Find $- f (x)$ .

Tap for more steps...

Step 4.1.1

Multiply $- x^{4} + x$ by $- 1$ .

$- f (x) = - (- x^{4} + x)$

Step 4.1.2

Apply the distributive property.

$- f (x) = x^{4} - x$

Step 4.2

Since $- x^{4} - x$ $\neq$ $x^{4} - x$ , the function is not odd.

The function is not odd

Step 5

The function is neither odd nor even

Step 6