Determine if Odd, Even, or Neither h(x)=x^4-x^12

$h (x) = x^{4} - x^{12}$

Step 1

Find $f (- x)$ .

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

$h (- x) = {(- x)}^{4} - {(- x)}^{12}$

Simplify each term.

Apply the product rule to $- x$ .

$h (- x) = {(- 1)}^{4} x^{4} - {(- x)}^{12}$

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

$h (- x) = 1 x^{4} - {(- x)}^{12}$

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

$h (- x) = x^{4} - {(- x)}^{12}$

Apply the product rule to $- x$ .

$h (- x) = x^{4} - ({(- 1)}^{12} x^{12})$

Multiply $- 1$ by ${(- 1)}^{12}$ by adding the exponents.

Move ${(- 1)}^{12}$ .

$h (- x) = x^{4} + {(- 1)}^{12} \cdot (- 1 x^{12})$

Multiply ${(- 1)}^{12}$ by $- 1$ .

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

$h (- x) = x^{4} + {(- 1)}^{12} \cdot ((- 1) x^{12})$

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

$h (- x) = x^{4} + {(- 1)}^{12 + 1} x^{12}$

Add $12$ and $1$ .

$h (- x) = x^{4} + {(- 1)}^{13} x^{12}$

Raise $- 1$ to the power of $13$ .

$h (- x) = x^{4} - x^{12}$

Step 2

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

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

Since $x^{4} - x^{12} = x^{4} - x^{12}$ , the function is even.

The function is even

Step 3