Determine if Odd, Even, or Neither 2x-|2x|

$2 x - | 2 x |$

Step 1

Write $2 x - | 2 x |$ as a function.

$f (x) = 2 x - | 2 x |$

Step 2

Simplify each term.

Remove non-negative terms from the absolute value.

$f (x) = 2 x - (2 | x |)$

Multiply $2$ by $- 1$ .

$f (x) = 2 x - 2 | x |$

Step 3

Find $f (- x)$ .

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

$f (- x) = 2 (- x) - 2 | - x |$

Simplify each term.

Multiply $- 1$ by $2$ .

$f (- x) = - 2 x - 2 | - x |$

Make $- x$ positive.

$f (- x) = - 2 x - 2 | x |$

Step 4

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

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

Since $- 2 x - 2 | x |$ $\neq$ $2 x - 2 | x |$ , the function is not even.

The function is not even

Step 5

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

Find $- f (x)$ .

Multiply $2 x - 2 | x |$ by $- 1$ .

$- f (x) = - (2 x - 2 | x |)$

Apply the distributive property.

$- f (x) = - (2 x) - (- 2 | x |)$

Multiply.

Multiply $2$ by $- 1$ .

$- f (x) = - 2 x - (- 2 | x |)$

Multiply $- 2$ by $- 1$ .

$- f (x) = - 2 x + 2 | x |$

Since $- 2 x - 2 | x |$ $\neq$ $- 2 x + 2 | x |$ , the function is not odd.

The function is not odd

Step 6

The function is neither odd nor even

Step 7