Determine if Odd, Even, or Neither f(x)=3x^5-x

$f (x) = 3 x^{5} - x$

Step 1

Find $f (- x)$ .

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

$f (- x) = 3 {(- x)}^{5} - (- x)$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = 3 ({(- 1)}^{5} x^{5}) - (- x)$

Raise $- 1$ to the power of $5$ .

$f (- x) = 3 (- x^{5}) - (- x)$

Multiply $- 1$ by $3$ .

$f (- x) = - 3 x^{5} - (- x)$

Multiply $- (- x)$ .

Multiply $- 1$ by $- 1$ .

$f (- x) = - 3 x^{5} + 1 x$

Multiply $x$ by $1$ .

$f (- x) = - 3 x^{5} + x$

Step 2

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

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

Since $- 3 x^{5} + x$ $\neq$ $3 x^{5} - x$ , the function is not even.

The function is not even

Step 3

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

Find $- f (x)$ .

Multiply $3 x^{5} - x$ by $- 1$ .

$- f (x) = - (3 x^{5} - x)$

Apply the distributive property.

$- f (x) = - (3 x^{5}) + x$

Multiply $3$ by $- 1$ .

$- f (x) = - 3 x^{5} + x$

Since $- 3 x^{5} + x = - 3 x^{5} + x$ , the function is odd.

The function is odd

Step 4