Determine if Odd, Even, or Neither f(x)=-3x^4-x^2

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

Step 1

Find $f (- x)$ .

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

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

Simplify each term.

Apply the product rule to $- x$ .

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

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

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

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

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

Apply the product rule to $- x$ .

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

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Move ${(- 1)}^{2}$ .

$f (- x) = - 3 x^{4} + {(- 1)}^{2} \cdot (- 1 x^{2})$

Multiply ${(- 1)}^{2}$ by $- 1$ .

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

$f (- x) = - 3 x^{4} + {(- 1)}^{2} \cdot ((- 1) x^{2})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $2$ and $1$ .

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

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

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

Step 2

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

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

Since $- 3 x^{4} - x^{2} = - 3 x^{4} - x^{2}$ , the function is even.

The function is even

Step 3