Determine if Odd, Even, or Neither p(x)=-|x|+12x^10+5

$p (x) = - | x | + 12 x^{10} + 5$

Step 1

Find $f (- x)$ .

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

$p (- x) = - | - x | + 12 {(- x)}^{10} + 5$

Simplify each term.

Make $- x$ positive.

$p (- x) = - | x | + 12 {(- x)}^{10} + 5$

Apply the product rule to $- x$ .

$p (- x) = - | x | + 12 ({(- 1)}^{10} x^{10}) + 5$

Raise $- 1$ to the power of $10$ .

$p (- x) = - | x | + 12 (1 x^{10}) + 5$

Multiply $x^{10}$ by $1$ .

$p (- x) = - | x | + 12 x^{10} + 5$

Step 2

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

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

Since $- | x | + 12 x^{10} + 5 = - | x | + 12 x^{10} + 5$ , the function is even.

The function is even

Step 3