Precalculus Examples

$y = x (4 - x^{2})$

Step 1

Write $y = x (4 - x^{2})$ as a function.

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

Step 2

Simplify.

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Step 2.1

Simplify by multiplying through.

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Step 2.1.1

Apply the distributive property.

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

Step 2.1.2

Reorder.

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Step 2.1.2.1

Move $4$ to the left of $x$ .

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

Step 2.1.2.2

Rewrite using the commutative property of multiplication.

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

Step 2.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.2.1

Move $x^{2}$ .

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

Step 2.2.2

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

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Step 2.2.2.1

Raise $x$ to the power of $1$ .

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

Step 2.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.3

Add $2$ and $1$ .

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

Step 3

Find $f (- x)$ .

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $- 1$ by $4$ .

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

Step 3.2.2

Apply the product rule to $- x$ .

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

Step 3.2.3

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

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Step 3.2.3.1

Move ${(- 1)}^{3}$ .

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

Step 3.2.3.2

Multiply ${(- 1)}^{3}$ by $- 1$ .

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Step 3.2.3.2.1

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

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

Step 3.2.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2.3.3

Add $3$ and $1$ .

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

Step 3.2.4

Raise $- 1$ to the power of $4$ .

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

Step 3.2.5

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

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

Step 4

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

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Step 4.1

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

Step 4.2

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

The function is not even

Step 5

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

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Step 5.1

Find $- f (x)$ .

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Step 5.1.1

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

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Multiply $4$ by $- 1$ .

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

Step 5.2

Since $- 4 x + x^{3} = - 4 x + x^{3}$ , the function is odd.

The function is odd

Step 6