Determine if Odd, Even, or Neither f(x)=x^2|x|+8

$f (x) = x^{2} | x | + 8$

Step 1

Find $f (- x)$ .

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

$f (- x) = {(- x)}^{2} | - x | + 8$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{2} x^{2} | - x | + 8$

Raise $- 1$ to the power of $2$ .

$f (- x) = 1 x^{2} | - x | + 8$

Multiply $x^{2}$ by $1$ .

$f (- x) = x^{2} | - x | + 8$

Make $- x$ positive.

$f (- x) = x^{2} | x | + 8$

Step 2

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

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

Since $x^{2} | x | + 8 = x^{2} | x | + 8$ , the function is even.

The function is even

Step 3