Determine if Odd, Even, or Neither f(x)=-7x^9+5x^2

$f (x) = - 7 x^{9} + 5 x^{2}$

Step 1

Find $f (- x)$ .

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

$f (- x) = - 7 {(- x)}^{9} + 5 {(- x)}^{2}$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = - 7 ({(- 1)}^{9} x^{9}) + 5 {(- x)}^{2}$

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

$f (- x) = - 7 (- x^{9}) + 5 {(- x)}^{2}$

Multiply $- 1$ by $- 7$ .

$f (- x) = 7 x^{9} + 5 {(- x)}^{2}$

Apply the product rule to $- x$ .

$f (- x) = 7 x^{9} + 5 ({(- 1)}^{2} x^{2})$

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

$f (- x) = 7 x^{9} + 5 (1 x^{2})$

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

$f (- x) = 7 x^{9} + 5 x^{2}$

Step 2

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

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

Since $7 x^{9} + 5 x^{2}$ $\neq$ $- 7 x^{9} + 5 x^{2}$ , the function is not even.

The function is not even

Step 3

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

Find $- f (x)$ .

Multiply $- 7 x^{9} + 5 x^{2}$ by $- 1$ .

$- f (x) = - (- 7 x^{9} + 5 x^{2})$

Apply the distributive property.

$- f (x) = - (- 7 x^{9}) - (5 x^{2})$

Multiply.

Multiply $- 7$ by $- 1$ .

$- f (x) = 7 x^{9} - (5 x^{2})$

Multiply $5$ by $- 1$ .

$- f (x) = 7 x^{9} - 5 x^{2}$

Since $7 x^{9} + 5 x^{2}$ $\neq$ $7 x^{9} - 5 x^{2}$ , the function is not odd.

The function is not odd

Step 4

The function is neither odd nor even

Step 5