Determine if Odd, Even, or Neither x^3-2x^2+3

$x^{3} - 2 x^{2} + 3$

Step 1

Write $x^{3} - 2 x^{2} + 3$ as a function.

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

Step 2

Find $f (- x)$ .

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

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

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{3} x^{3} - 2 {(- x)}^{2} + 3$

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

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

Apply the product rule to $- x$ .

$f (- x) = - x^{3} - 2 ({(- 1)}^{2} x^{2}) + 3$

Raise $- 1$ to the power of $2$ .

$f (- x) = - x^{3} - 2 (1 x^{2}) + 3$

Multiply $x^{2}$ by $1$ .

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

Step 3

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

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

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

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

Apply the distributive property.

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

Simplify.

Multiply $- 2$ by $- 1$ .

$- f (x) = - x^{3} + 2 x^{2} - 1 \cdot 3$

Multiply $- 1$ by $3$ .

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

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

The function is not odd

Step 5

The function is neither odd nor even

Step 6