Precalculus Examples

$f (x) = x^{4} + 5 x^{2} - x^{5}$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$f (- x) = {(- x)}^{4} + 5 {(- x)}^{2} - {(- x)}^{5}$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{4} x^{4} + 5 {(- x)}^{2} - {(- x)}^{5}$

Step 1.2.2

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

$f (- x) = 1 x^{4} + 5 {(- x)}^{2} - {(- x)}^{5}$

Step 1.2.3

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

$f (- x) = x^{4} + 5 {(- x)}^{2} - {(- x)}^{5}$

Step 1.2.4

Apply the product rule to $- x$ .

$f (- x) = x^{4} + 5 ({(- 1)}^{2} x^{2}) - {(- x)}^{5}$

Step 1.2.5

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

$f (- x) = x^{4} + 5 (1 x^{2}) - {(- x)}^{5}$

Step 1.2.6

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

$f (- x) = x^{4} + 5 x^{2} - {(- x)}^{5}$

Step 1.2.7

Apply the product rule to $- x$ .

$f (- x) = x^{4} + 5 x^{2} - ({(- 1)}^{5} x^{5})$

Step 1.2.8

Multiply $- 1$ by ${(- 1)}^{5}$ by adding the exponents.

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Step 1.2.8.1

Move ${(- 1)}^{5}$ .

$f (- x) = x^{4} + 5 x^{2} + {(- 1)}^{5} \cdot (- 1 x^{5})$

Step 1.2.8.2

Multiply ${(- 1)}^{5}$ by $- 1$ .

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Step 1.2.8.2.1

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

$f (- x) = x^{4} + 5 x^{2} + {(- 1)}^{5} \cdot ((- 1) x^{5})$

Step 1.2.8.2.2

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

$f (- x) = x^{4} + 5 x^{2} + {(- 1)}^{5 + 1} x^{5}$

Step 1.2.8.3

Add $5$ and $1$ .

$f (- x) = x^{4} + 5 x^{2} + {(- 1)}^{6} x^{5}$

Step 1.2.9

Raise $- 1$ to the power of $6$ .

$f (- x) = x^{4} + 5 x^{2} + 1 x^{5}$

Step 1.2.10

Multiply $x^{5}$ by $1$ .

$f (- x) = x^{4} + 5 x^{2} + x^{5}$

Step 2

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

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Step 2.1

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

Step 2.2

Since $x^{4} + 5 x^{2} + x^{5}$ $\neq$ $x^{4} + 5 x^{2} - x^{5}$ , the function is not even.

The function is not even

Step 3

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

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Step 3.1

Find $- f (x)$ .

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Step 3.1.1

Multiply $x^{4} + 5 x^{2} - x^{5}$ by $- 1$ .

$- f (x) = - (x^{4} + 5 x^{2} - x^{5})$

Step 3.1.2

Apply the distributive property.

$- f (x) = - x^{4} - (5 x^{2}) + x^{5}$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Multiply $5$ by $- 1$ .

$- f (x) = - x^{4} - 5 x^{2} + x^{5}$

Step 3.1.3.2

Multiply $- - x^{5}$ .

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Step 3.1.3.2.1

Multiply $- 1$ by $- 1$ .

$- f (x) = - x^{4} - 5 x^{2} + 1 x^{5}$

Step 3.1.3.2.2

Multiply $x^{5}$ by $1$ .

$- f (x) = - x^{4} - 5 x^{2} + x^{5}$

Step 3.2

Since $x^{4} + 5 x^{2} + x^{5}$ $\neq$ $- x^{4} - 5 x^{2} + x^{5}$ , the function is not odd.

The function is not odd

Step 4

The function is neither odd nor even

Step 5