Determine if Odd, Even, or Neither -10x^11+x^9-x^3+7x

$- 10 x^{11} + x^{9} - x^{3} + 7 x$

Step 1

Write $- 10 x^{11} + x^{9} - x^{3} + 7 x$ as a function.

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

Step 2

Find $f (- x)$ .

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

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

Simplify each term.

Apply the product rule to $- x$ .

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

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

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

Multiply $- 1$ by $- 10$ .

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

Apply the product rule to $- x$ .

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

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

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

Apply the product rule to $- x$ .

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

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

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

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

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

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

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

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

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

Add $3$ and $1$ .

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

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

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

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

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

Multiply $- 1$ by $7$ .

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

Step 3

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

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

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 4

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

Find $- f (x)$ .

Multiply $- 10 x^{11} + x^{9} - x^{3} + 7 x$ by $- 1$ .

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

Apply the distributive property.

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

Simplify.

Multiply $- 10$ by $- 1$ .

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

Multiply $- - x^{3}$ .

Multiply $- 1$ by $- 1$ .

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

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

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

Multiply $7$ by $- 1$ .

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

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 5

Determine if Odd, Even, or Neither -10x^11+x^9-x^3+7x

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