Determine if Odd, Even, or Neither y=-3x^4-2x-5

$y = - 3 x^{4} - 2 x - 5$

Step 1

Write $y = - 3 x^{4} - 2 x - 5$ as a function.

$f (x) = - 3 x^{4} - 2 x - 5$

Step 2

Find $f (- x)$ .

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

$f (- x) = - 3 {(- x)}^{4} - 2 (- x) - 5$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = - 3 ({(- 1)}^{4} x^{4}) - 2 (- x) - 5$

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

$f (- x) = - 3 (1 x^{4}) - 2 (- x) - 5$

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

$f (- x) = - 3 x^{4} - 2 (- x) - 5$

Multiply $- 1$ by $- 2$ .

$f (- x) = - 3 x^{4} + 2 x - 5$

Step 3

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

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

Since $- 3 x^{4} + 2 x - 5$ $\neq$ $- 3 x^{4} - 2 x - 5$ , 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 $- 3 x^{4} - 2 x - 5$ by $- 1$ .

$- f (x) = - (- 3 x^{4} - 2 x - 5)$

Apply the distributive property.

$- f (x) = - (- 3 x^{4}) - (- 2 x) + 5$

Simplify.

Multiply $- 3$ by $- 1$ .

$- f (x) = 3 x^{4} - (- 2 x) + 5$

Multiply $- 2$ by $- 1$ .

$- f (x) = 3 x^{4} + 2 x + 5$

Multiply $- 1$ by $- 5$ .

$- f (x) = 3 x^{4} + 2 x + 5$

Since $- 3 x^{4} + 2 x - 5$ $\neq$ $3 x^{4} + 2 x + 5$ , the function is not odd.

The function is not odd

Step 5

The function is neither odd nor even

Step 6