Calculus Examples

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

Step 1

Set the denominator in $\frac{x^{3}}{x^{2} - 1}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 1 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $1$ to both sides of the equation.

$x^{2} = 1$

Step 2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1}$

Step 2.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Step 2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

Step 2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 1) \cup (- 1, 1) \cup (1, \infty)$

Set-Builder Notation:

${x | x \neq 1, - 1}$

Step 4

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

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 5

Determine the domain and range.

Domain: $(- \infty, - 1) \cup (- 1, 1) \cup (1, \infty), {x | x \neq 1, - 1}$

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

Step 6