Calculus Examples

$f (x) = \frac{2}{x} - \frac{8}{x^{2}}$

Step 1

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

$x = 0$

Step 2

Set the denominator in $\frac{8}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 3

Solve for $x$ .

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Step 3.1

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

$x = \pm \sqrt{0}$

Step 3.2

Simplify $\pm \sqrt{0}$ .

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Step 3.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 3.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 3.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 5

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

Interval Notation:

$(- \infty, \frac{1}{8}]$

Set-Builder Notation:

${y | y \leq \frac{1}{8}}$

Step 6

Determine the domain and range.

Domain: $(- \infty, 0) \cup (0, \infty), {x | x \neq 0}$

Range: $(- \infty, \frac{1}{8}], {y | y \leq \frac{1}{8}}$

Step 7