Algebra Examples

$h (x) = - 10 x^{11} + x^{9} - x^{3} + 7 x$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$h (- x) = - 10 {(- x)}^{11} + {(- x)}^{9} - {(- x)}^{3} + 7 (- x)$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $- x$ .

$h (- x) = - 10 ({(- 1)}^{11} x^{11}) + {(- x)}^{9} - {(- x)}^{3} + 7 (- x)$

Step 1.2.2

Raise $- 1$ to the power of $11$ .

$h (- x) = - 10 (- x^{11}) + {(- x)}^{9} - {(- x)}^{3} + 7 (- x)$

Step 1.2.3

Multiply $- 1$ by $- 10$ .

$h (- x) = 10 x^{11} + {(- x)}^{9} - {(- x)}^{3} + 7 (- x)$

Step 1.2.4

Apply the product rule to $- x$ .

$h (- x) = 10 x^{11} + {(- 1)}^{9} x^{9} - {(- x)}^{3} + 7 (- x)$

Step 1.2.5

Raise $- 1$ to the power of $9$ .

$h (- x) = 10 x^{11} - x^{9} - {(- x)}^{3} + 7 (- x)$

Step 1.2.6

Apply the product rule to $- x$ .

$h (- x) = 10 x^{11} - x^{9} - ({(- 1)}^{3} x^{3}) + 7 (- x)$

Step 1.2.7

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

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Step 1.2.7.1

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

$h (- x) = 10 x^{11} - x^{9} + {(- 1)}^{3} \cdot (- 1 x^{3}) + 7 (- x)$

Step 1.2.7.2

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

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Step 1.2.7.2.1

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

$h (- x) = 10 x^{11} - x^{9} + {(- 1)}^{3} \cdot ((- 1) x^{3}) + 7 (- x)$

Step 1.2.7.2.2

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

$h (- x) = 10 x^{11} - x^{9} + {(- 1)}^{3 + 1} x^{3} + 7 (- x)$

Step 1.2.7.3

Add $3$ and $1$ .

$h (- x) = 10 x^{11} - x^{9} + {(- 1)}^{4} x^{3} + 7 (- x)$

Step 1.2.8

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

$h (- x) = 10 x^{11} - x^{9} + 1 x^{3} + 7 (- x)$

Step 1.2.9

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

$h (- x) = 10 x^{11} - x^{9} + x^{3} + 7 (- x)$

Step 1.2.10

Multiply $- 1$ by $7$ .

$h (- x) = 10 x^{11} - x^{9} + x^{3} - 7 x$

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 $10 x^{11} - x^{9} + x^{3} - 7 x$ $\neq$ $- 10 x^{11} + x^{9} - x^{3} + 7 x$ , 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 $- 10 x^{11} + x^{9} - x^{3} + 7 x$ by $- 1$ .

$- f (x) = - (- 10 x^{11} + x^{9} - x^{3} + 7 x)$

Step 3.1.2

Apply the distributive property.

$- f (x) = - (- 10 x^{11}) - x^{9} + x^{3} - (7 x)$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Multiply $- 10$ by $- 1$ .

$- f (x) = 10 x^{11} - x^{9} + x^{3} - (7 x)$

Step 3.1.3.2

Multiply $- - x^{3}$ .

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Step 3.1.3.2.1

Multiply $- 1$ by $- 1$ .

$- f (x) = 10 x^{11} - x^{9} + 1 x^{3} - (7 x)$

Step 3.1.3.2.2

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

$- f (x) = 10 x^{11} - x^{9} + x^{3} - (7 x)$

Step 3.1.3.3

Multiply $7$ by $- 1$ .

$- f (x) = 10 x^{11} - x^{9} + x^{3} - 7 x$

Step 3.2

Since $10 x^{11} - x^{9} + x^{3} - 7 x = 10 x^{11} - x^{9} + x^{3} - 7 x$ , the function is odd.

The function is odd

Step 4