Determine if Odd, Even, or Neither (x^3)/2-5

$\frac{x^{3}}{2} - 5$

Step 1

Write $\frac{x^{3}}{2} - 5$ as a function.

$f (x) = \frac{x^{3}}{2} - 5$

Step 2

Find $f (- x)$ .

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

$f (- x) = \frac{{(- x)}^{3}}{2} - 5$

Simplify each term.

Simplify the numerator.

Apply the product rule to $- x$ .

$f (- x) = \frac{{(- 1)}^{3} x^{3}}{2} - 5$

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

$f (- x) = \frac{- x^{3}}{2} - 5$

Move the negative in front of the fraction.

$f (- x) = - \frac{x^{3}}{2} - 5$

Step 3

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

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

Since $- \frac{x^{3}}{2} - 5$ $\neq$ $\frac{x^{3}}{2} - 5$ , 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 $\frac{x^{3}}{2} - 5$ by $- 1$ .

$- f (x) = - (\frac{x^{3}}{2} - 5)$

Apply the distributive property.

$- f (x) = - \frac{x^{3}}{2} + 5$

Since $- \frac{x^{3}}{2} - 5$ $\neq$ $- \frac{x^{3}}{2} + 5$ , the function is not odd.

The function is not odd

Step 5

The function is neither odd nor even

Step 6