Determine if Odd, Even, or Neither -7x^6+3x^4-9x^2+8

$- 7 x^{6} + 3 x^{4} - 9 x^{2} + 8$

Step 1

Write $- 7 x^{6} + 3 x^{4} - 9 x^{2} + 8$ as a function.

$f (x) = - 7 x^{6} + 3 x^{4} - 9 x^{2} + 8$

Step 2

Find $f (- x)$ .

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

$f (- x) = - 7 {(- x)}^{6} + 3 {(- x)}^{4} - 9 {(- x)}^{2} + 8$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = - 7 ({(- 1)}^{6} x^{6}) + 3 {(- x)}^{4} - 9 {(- x)}^{2} + 8$

Raise $- 1$ to the power of $6$ .

$f (- x) = - 7 (1 x^{6}) + 3 {(- x)}^{4} - 9 {(- x)}^{2} + 8$

Multiply $x^{6}$ by $1$ .

$f (- x) = - 7 x^{6} + 3 {(- x)}^{4} - 9 {(- x)}^{2} + 8$

Apply the product rule to $- x$ .

$f (- x) = - 7 x^{6} + 3 ({(- 1)}^{4} x^{4}) - 9 {(- x)}^{2} + 8$

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

$f (- x) = - 7 x^{6} + 3 (1 x^{4}) - 9 {(- x)}^{2} + 8$

Multiply $x^{4}$ by $1$ .

$f (- x) = - 7 x^{6} + 3 x^{4} - 9 {(- x)}^{2} + 8$

Apply the product rule to $- x$ .

$f (- x) = - 7 x^{6} + 3 x^{4} - 9 ({(- 1)}^{2} x^{2}) + 8$

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

$f (- x) = - 7 x^{6} + 3 x^{4} - 9 (1 x^{2}) + 8$

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

$f (- x) = - 7 x^{6} + 3 x^{4} - 9 x^{2} + 8$

Step 3

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

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

Since $- 7 x^{6} + 3 x^{4} - 9 x^{2} + 8 = - 7 x^{6} + 3 x^{4} - 9 x^{2} + 8$ , the function is even.

The function is even

Step 4