Algebra Examples

$- x^{7} + 9 x^{3} + 3 x$

Step 1

Write $- x^{7} + 9 x^{3} + 3 x$ as a function.

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

Step 2

Find $f (- x)$ .

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Step 2.1

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

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

Step 2.2

Simplify each term.

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Step 2.2.1

Apply the product rule to $- x$ .

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

Step 2.2.2

Multiply $- 1$ by ${(- 1)}^{7}$ by adding the exponents.

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Step 2.2.2.1

Move ${(- 1)}^{7}$ .

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

Step 2.2.2.2

Multiply ${(- 1)}^{7}$ by $- 1$ .

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Step 2.2.2.2.1

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

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

Step 2.2.2.2.2

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

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

Step 2.2.2.3

Add $7$ and $1$ .

$f (- x) = {(- 1)}^{8} x^{7} + 9 {(- x)}^{3} + 3 (- x)$

Step 2.2.3

Raise $- 1$ to the power of $8$ .

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

Step 2.2.4

Multiply $x^{7}$ by $1$ .

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

Step 2.2.5

Apply the product rule to $- x$ .

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

Step 2.2.6

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

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

Step 2.2.7

Multiply $- 1$ by $9$ .

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

Step 2.2.8

Multiply $- 1$ by $3$ .

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

Step 3

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

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Step 3.1

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

Step 3.2

Since $x^{7} - 9 x^{3} - 3 x$ $\neq$ $- x^{7} + 9 x^{3} + 3 x$ , the function is not even.

The function is not even

Step 4

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

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Step 4.1

Find $- f (x)$ .

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Step 4.1.1

Multiply $- x^{7} + 9 x^{3} + 3 x$ by $- 1$ .

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Simplify.

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Step 4.1.3.1

Multiply $9$ by $- 1$ .

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

Step 4.1.3.2

Multiply $3$ by $- 1$ .

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

Step 4.2

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

The function is odd

Step 5