Determine if Odd, Even, or Neither -6x^5+3x^2

$- 6 x^{5} + 3 x^{2}$

Step 1

Write $- 6 x^{5} + 3 x^{2}$ as a function.

$f (x) = - 6 x^{5} + 3 x^{2}$

Step 2

Find $f (- x)$ .

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

$f (- x) = - 6 {(- x)}^{5} + 3 {(- x)}^{2}$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = - 6 ({(- 1)}^{5} x^{5}) + 3 {(- x)}^{2}$

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

$f (- x) = - 6 (- x^{5}) + 3 {(- x)}^{2}$

Multiply $- 1$ by $- 6$ .

$f (- x) = 6 x^{5} + 3 {(- x)}^{2}$

Apply the product rule to $- x$ .

$f (- x) = 6 x^{5} + 3 ({(- 1)}^{2} x^{2})$

Raise $- 1$ to the power of $2$ .

$f (- x) = 6 x^{5} + 3 (1 x^{2})$

Multiply $x^{2}$ by $1$ .

$f (- x) = 6 x^{5} + 3 x^{2}$

Step 3

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

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

Since $6 x^{5} + 3 x^{2}$ $\neq$ $- 6 x^{5} + 3 x^{2}$ , 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 $- 6 x^{5} + 3 x^{2}$ by $- 1$ .

$- f (x) = - (- 6 x^{5} + 3 x^{2})$

Apply the distributive property.

$- f (x) = - (- 6 x^{5}) - (3 x^{2})$

Multiply.

Multiply $- 6$ by $- 1$ .

$- f (x) = 6 x^{5} - (3 x^{2})$

Multiply $3$ by $- 1$ .

$- f (x) = 6 x^{5} - 3 x^{2}$

Since $6 x^{5} + 3 x^{2}$ $\neq$ $6 x^{5} - 3 x^{2}$ , the function is not odd.

The function is not odd

Step 5

The function is neither odd nor even

Step 6