Determine if Odd, Even, or Neither -6x^5+7x^3

$- 6 x^{5} + 7 x^{3}$

Step 1

Write $- 6 x^{5} + 7 x^{3}$ as a function.

$f (x) = - 6 x^{5} + 7 x^{3}$

Step 2

Find $f (- x)$ .

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

$f (- x) = - 6 {(- x)}^{5} + 7 {(- x)}^{3}$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = - 6 ({(- 1)}^{5} x^{5}) + 7 {(- x)}^{3}$

Raise $- 1$ to the power of $5$ .

$f (- x) = - 6 (- x^{5}) + 7 {(- x)}^{3}$

Multiply $- 1$ by $- 6$ .

$f (- x) = 6 x^{5} + 7 {(- x)}^{3}$

Apply the product rule to $- x$ .

$f (- x) = 6 x^{5} + 7 ({(- 1)}^{3} x^{3})$

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

$f (- x) = 6 x^{5} + 7 (- x^{3})$

Multiply $- 1$ by $7$ .

$f (- x) = 6 x^{5} - 7 x^{3}$

Step 3

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

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

Since $6 x^{5} - 7 x^{3}$ $\neq$ $- 6 x^{5} + 7 x^{3}$ , 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 $- 6 x^{5} + 7 x^{3}$ by $- 1$ .

$- f (x) = - (- 6 x^{5} + 7 x^{3})$

Apply the distributive property.

$- f (x) = - (- 6 x^{5}) - (7 x^{3})$

Multiply.

Multiply $- 6$ by $- 1$ .

$- f (x) = 6 x^{5} - (7 x^{3})$

Multiply $7$ by $- 1$ .

$- f (x) = 6 x^{5} - 7 x^{3}$

Since $6 x^{5} - 7 x^{3} = 6 x^{5} - 7 x^{3}$ , the function is odd.

The function is odd

Step 5