Determine if Odd, Even, or Neither f(x)=x^6-x^2

$f (x) = x^{6} - x^{2}$

Step 1

Find $f (- x)$ .

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

$f (- x) = {(- x)}^{6} - {(- x)}^{2}$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{6} x^{6} - {(- x)}^{2}$

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

$f (- x) = 1 x^{6} - {(- x)}^{2}$

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

$f (- x) = x^{6} - {(- x)}^{2}$

Apply the product rule to $- x$ .

$f (- x) = x^{6} - ({(- 1)}^{2} x^{2})$

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

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

$f (- x) = x^{6} + {(- 1)}^{2} \cdot (- 1 x^{2})$

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

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

$f (- x) = x^{6} + {(- 1)}^{2} \cdot ((- 1) x^{2})$

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

$f (- x) = x^{6} + {(- 1)}^{2 + 1} x^{2}$

Add $2$ and $1$ .

$f (- x) = x^{6} + {(- 1)}^{3} x^{2}$

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

$f (- x) = x^{6} - x^{2}$

Step 2

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

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

Since $x^{6} - x^{2} = x^{6} - x^{2}$ , the function is even.

The function is even

Step 3