Precalculus Examples

$f (x) = | x^{4} - x^{3} |$

Step 1

Find $f (- x)$ .

Tap for more steps...

Step 1.1

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

$f (- x) = | {(- x)}^{4} - {(- x)}^{3} |$

Step 1.2

Simplify each term.

Tap for more steps...

Step 1.2.1

Apply the product rule to $- x$ .

$f (- x) = | {(- 1)}^{4} x^{4} - {(- x)}^{3} |$

Step 1.2.2

Raise $- 1$ to the power of $4$ .

$f (- x) = | 1 x^{4} - {(- x)}^{3} |$

Step 1.2.3

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

$f (- x) = | x^{4} - {(- x)}^{3} |$

Step 1.2.4

Apply the product rule to $- x$ .

$f (- x) = | x^{4} - ({(- 1)}^{3} x^{3}) |$

Step 1.2.5

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

Tap for more steps...

Step 1.2.5.1

Move ${(- 1)}^{3}$ .

$f (- x) = | x^{4} + {(- 1)}^{3} \cdot (- 1 x^{3}) |$

Step 1.2.5.2

Multiply ${(- 1)}^{3}$ by $- 1$ .

Tap for more steps...

Step 1.2.5.2.1

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

$f (- x) = | x^{4} + {(- 1)}^{3} \cdot ((- 1) x^{3}) |$

Step 1.2.5.2.2

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

$f (- x) = | x^{4} + {(- 1)}^{3 + 1} x^{3} |$

Step 1.2.5.3

Add $3$ and $1$ .

$f (- x) = | x^{4} + {(- 1)}^{4} x^{3} |$

Step 1.2.6

Raise $- 1$ to the power of $4$ .

$f (- x) = | x^{4} + 1 x^{3} |$

Step 1.2.7

Multiply $x^{3}$ by $1$ .

$f (- x) = | x^{4} + x^{3} |$

Step 2

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

Tap for more steps...

Step 2.1

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

Step 2.2

Since $| x^{4} + x^{3} |$ $\neq$ $| x^{4} - x^{3} |$ , the function is not even.

The function is not even

Step 3

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

Tap for more steps...

Step 3.1

Multiply $- 1$ by $| x^{4} - x^{3} |$ .

$- f (x) = - | x^{4} - x^{3} |$

Step 3.2

Since $| x^{4} + x^{3} |$ $\neq$ $- | x^{4} - x^{3} |$ , the function is not odd.

The function is not odd

Step 4

The function is neither odd nor even

Step 5