Determine if Odd, Even, or Neither -3x^7+8x^3-5x

$- 3 x^{7} + 8 x^{3} - 5 x$

Step 1

Write $- 3 x^{7} + 8 x^{3} - 5 x$ as a function.

$f (x) = - 3 x^{7} + 8 x^{3} - 5 x$

Step 2

Find $f (- x)$ .

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

$f (- x) = - 3 {(- x)}^{7} + 8 {(- x)}^{3} - 5 (- x)$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = - 3 ({(- 1)}^{7} x^{7}) + 8 {(- x)}^{3} - 5 (- x)$

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

$f (- x) = - 3 (- x^{7}) + 8 {(- x)}^{3} - 5 (- x)$

Multiply $- 1$ by $- 3$ .

$f (- x) = 3 x^{7} + 8 {(- x)}^{3} - 5 (- x)$

Apply the product rule to $- x$ .

$f (- x) = 3 x^{7} + 8 ({(- 1)}^{3} x^{3}) - 5 (- x)$

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

$f (- x) = 3 x^{7} + 8 (- x^{3}) - 5 (- x)$

Multiply $- 1$ by $8$ .

$f (- x) = 3 x^{7} - 8 x^{3} - 5 (- x)$

Multiply $- 1$ by $- 5$ .

$f (- x) = 3 x^{7} - 8 x^{3} + 5 x$

Step 3

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

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

Since $3 x^{7} - 8 x^{3} + 5 x$ $\neq$ $- 3 x^{7} + 8 x^{3} - 5 x$ , 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 $- 3 x^{7} + 8 x^{3} - 5 x$ by $- 1$ .

$- f (x) = - (- 3 x^{7} + 8 x^{3} - 5 x)$

Apply the distributive property.

$- f (x) = - (- 3 x^{7}) - (8 x^{3}) - (- 5 x)$

Simplify.

Multiply $- 3$ by $- 1$ .

$- f (x) = 3 x^{7} - (8 x^{3}) - (- 5 x)$

Multiply $8$ by $- 1$ .

$- f (x) = 3 x^{7} - 8 x^{3} - (- 5 x)$

Multiply $- 5$ by $- 1$ .

$- f (x) = 3 x^{7} - 8 x^{3} + 5 x$

Since $3 x^{7} - 8 x^{3} + 5 x = 3 x^{7} - 8 x^{3} + 5 x$ , the function is odd.

The function is odd

Step 5