Find the Domain and Range (x)=x^3

$(x) = x^{3}$

Subtract $x^{3}$ from both sides of the equation.

$x - x^{3} = 0$

Factor the left side of the equation.

Factor $x$ out of $x - x^{3}$ .

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

$x - x^{3} = 0$

Factor $x$ out of $x^{1}$ .

$x \cdot 1 - x^{3} = 0$

Factor $x$ out of $- x^{3}$ .

$x \cdot 1 + x (- x^{2}) = 0$

Factor $x$ out of $x \cdot 1 + x (- x^{2})$ .

$x (1 - x^{2}) = 0$

Rewrite $1$ as $1^{2}$ .

$x (1^{2} - x^{2}) = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$x ((1 + x) (1 - x)) = 0$

Remove unnecessary parentheses.

$x (1 + x) (1 - x) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$1 + x = 0$

$1 - x = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $1 + x$ equal to $0$ and solve for $x$ .

Set $1 + x$ equal to $0$ .

$1 + x = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $1 - x$ equal to $0$ and solve for $x$ .

Set $1 - x$ equal to $0$ .

$1 - x = 0$

Solve $1 - x = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$- x = - 1$

Divide each term in $- x = - 1$ by $- 1$ and simplify.

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Simplify the right side.

Divide $- 1$ by $- 1$ .

$x = 1$

The final solution is all the values that make $x (1 + x) (1 - x) = 0$ true.

$x = 0, - 1, 1$

The domain is the set of all valid $x$ values.

${x | x = 0, - 1, 1}$

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Determine the domain and range.

Domain: ${x | x = 0, - 1, 1}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Find the Domain and Range (x)=x^3

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