Determine if Odd, Even, or Neither f(x)=4x-|4x|

$f (x) = 4 x - | 4 x |$

Step 1

Simplify each term.

Remove non-negative terms from the absolute value.

$f (x) = 4 x - (4 | x |)$

Multiply $4$ by $- 1$ .

$f (x) = 4 x - 4 | x |$

Step 2

Find $f (- x)$ .

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

$f (- x) = 4 (- x) - 4 | - x |$

Simplify each term.

Multiply $- 1$ by $4$ .

$f (- x) = - 4 x - 4 | - x |$

Make $- x$ positive.

$f (- x) = - 4 x - 4 | x |$

Step 3

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

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

Since $- 4 x - 4 | x |$ $\neq$ $4 x - 4 | x |$ , the function is not even.

The function is not even

Step 4

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

Find $- f (x)$ .

Multiply $4 x - 4 | x |$ by $- 1$ .

$- f (x) = - (4 x - 4 | x |)$

Apply the distributive property.

$- f (x) = - (4 x) - (- 4 | x |)$

Multiply.

Multiply $4$ by $- 1$ .

$- f (x) = - 4 x - (- 4 | x |)$

Multiply $- 4$ by $- 1$ .

$- f (x) = - 4 x + 4 | x |$

Since $- 4 x - 4 | x |$ $\neq$ $- 4 x + 4 | x |$ , the function is not odd.

The function is not odd

Step 5

The function is neither odd nor even

Step 6