Precalculus Examples

$f (x) = 4 x^{3} - 3 x^{2} + 5$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$f (- x) = 4 {(- x)}^{3} - 3 {(- x)}^{2} + 5$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $- x$ .

$f (- x) = 4 ({(- 1)}^{3} x^{3}) - 3 {(- x)}^{2} + 5$

Step 1.2.2

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

$f (- x) = 4 (- x^{3}) - 3 {(- x)}^{2} + 5$

Step 1.2.3

Multiply $- 1$ by $4$ .

$f (- x) = - 4 x^{3} - 3 {(- x)}^{2} + 5$

Step 1.2.4

Apply the product rule to $- x$ .

$f (- x) = - 4 x^{3} - 3 ({(- 1)}^{2} x^{2}) + 5$

Step 1.2.5

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

$f (- x) = - 4 x^{3} - 3 (1 x^{2}) + 5$

Step 1.2.6

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

$f (- x) = - 4 x^{3} - 3 x^{2} + 5$

Step 2

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

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Step 2.1

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

Step 2.2

Since $- 4 x^{3} - 3 x^{2} + 5$ $\neq$ $4 x^{3} - 3 x^{2} + 5$ , the function is not even.

The function is not even

Step 3

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

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Step 3.1

Find $- f (x)$ .

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Step 3.1.1

Multiply $4 x^{3} - 3 x^{2} + 5$ by $- 1$ .

$- f (x) = - (4 x^{3} - 3 x^{2} + 5)$

Step 3.1.2

Apply the distributive property.

$- f (x) = - (4 x^{3}) - (- 3 x^{2}) - 1 \cdot 5$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Multiply $4$ by $- 1$ .

$- f (x) = - 4 x^{3} - (- 3 x^{2}) - 1 \cdot 5$

Step 3.1.3.2

Multiply $- 3$ by $- 1$ .

$- f (x) = - 4 x^{3} + 3 x^{2} - 1 \cdot 5$

Step 3.1.3.3

Multiply $- 1$ by $5$ .

$- f (x) = - 4 x^{3} + 3 x^{2} - 5$

Step 3.2

Since $- 4 x^{3} - 3 x^{2} + 5$ $\neq$ $- 4 x^{3} + 3 x^{2} - 5$ , the function is not odd.

The function is not odd

Step 4

The function is neither odd nor even

Step 5