Precalculus Examples

$f (x) = x^{3} + x^{7} - x^{9}$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$f (- x) = {(- x)}^{3} + {(- x)}^{7} - {(- x)}^{9}$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{3} x^{3} + {(- x)}^{7} - {(- x)}^{9}$

Step 1.2.2

Raise $- 1$ to the power of $3$ .

$f (- x) = - x^{3} + {(- x)}^{7} - {(- x)}^{9}$

Step 1.2.3

Apply the product rule to $- x$ .

$f (- x) = - x^{3} + {(- 1)}^{7} x^{7} - {(- x)}^{9}$

Step 1.2.4

Raise $- 1$ to the power of $7$ .

$f (- x) = - x^{3} - x^{7} - {(- x)}^{9}$

Step 1.2.5

Apply the product rule to $- x$ .

$f (- x) = - x^{3} - x^{7} - ({(- 1)}^{9} x^{9})$

Step 1.2.6

Multiply $- 1$ by ${(- 1)}^{9}$ by adding the exponents.

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Step 1.2.6.1

Move ${(- 1)}^{9}$ .

$f (- x) = - x^{3} - x^{7} + {(- 1)}^{9} \cdot (- 1 x^{9})$

Step 1.2.6.2

Multiply ${(- 1)}^{9}$ by $- 1$ .

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Step 1.2.6.2.1

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

$f (- x) = - x^{3} - x^{7} + {(- 1)}^{9} \cdot ((- 1) x^{9})$

Step 1.2.6.2.2

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

$f (- x) = - x^{3} - x^{7} + {(- 1)}^{9 + 1} x^{9}$

Step 1.2.6.3

Add $9$ and $1$ .

$f (- x) = - x^{3} - x^{7} + {(- 1)}^{10} x^{9}$

Step 1.2.7

Raise $- 1$ to the power of $10$ .

$f (- x) = - x^{3} - x^{7} + 1 x^{9}$

Step 1.2.8

Multiply $x^{9}$ by $1$ .

$f (- x) = - x^{3} - x^{7} + x^{9}$

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 $- x^{3} - x^{7} + x^{9}$ $\neq$ $x^{3} + x^{7} - x^{9}$ , 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 $x^{3} + x^{7} - x^{9}$ by $- 1$ .

$- f (x) = - (x^{3} + x^{7} - x^{9})$

Step 3.1.2

Apply the distributive property.

$- f (x) = - x^{3} - x^{7} + x^{9}$

Step 3.2

Since $- x^{3} - x^{7} + x^{9} = - x^{3} - x^{7} + x^{9}$ , the function is odd.

The function is odd

Step 4