Determine if Odd, Even, or Neither 3x^3+5

$3 x^{3} + 5$

Step 1

Write $3 x^{3} + 5$ as a function.

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

Step 2

Find $f (- x)$ .

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

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

Simplify each term.

Apply the product rule to $- x$ .

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

Raise $- 1$ to the power of $3$ .

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

Multiply $- 1$ by $3$ .

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

Step 3

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

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

Since $- 3 x^{3} + 5$ $\neq$ $3 x^{3} + 5$ , 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 $3 x^{3} + 5$ by $- 1$ .

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

Apply the distributive property.

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

Multiply.

Multiply $3$ by $- 1$ .

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

Multiply $- 1$ by $5$ .

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

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

The function is not odd

Step 5

The function is neither odd nor even

Step 6