Calculus Examples

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

Step 1

Remove parentheses.

$\frac{- 1}{2 x^{2}}$

Step 2

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

$2 x^{2} = 0$

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $2 x^{2} = 0$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 x^{2} = 0$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{0}{2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $2$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{0}{2}$

Step 3.1.2.1.2

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

$x^{2} = \frac{0}{2}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $0$ by $2$ .

$x^{2} = 0$

Step 3.2

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

$x = \pm \sqrt{0}$

Step 3.3

Simplify $\pm \sqrt{0}$ .

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Step 3.3.1

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

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

Step 3.3.2

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

$x = \pm 0$

Step 3.3.3

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:

$(- \infty, 0)$

Set-Builder Notation:

${y | y < 0}$

Step 6

Determine the domain and range.

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

Range: $(- \infty, 0), {y | y < 0}$

Step 7