Precalculus Examples

$f (x) = \frac{3}{x^{2} - 4}$

Step 1

Set the denominator in $\frac{3}{x^{2} - 4}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 4 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $4$ to both sides of the equation.

$x^{2} = 4$

Step 2.2

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

$x = \pm \sqrt{4}$

Step 2.3

Simplify $\pm \sqrt{4}$ .

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Step 2.3.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.3.2

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

$x = \pm 2$

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.

$x = 2$

Step 2.4.2

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

$x = - 2$

Step 2.4.3

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

$x = 2, - 2$

Step 3

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

Interval Notation:

$(- \infty, - 2) \cup (- 2, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 2, - 2}$

Step 4

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

Interval Notation:

$(- \infty, - \frac{3}{4}] \cup (0, \infty)$

Set-Builder Notation:

${y | y \leq - \frac{3}{4}, y > 0}$

Step 5

Determine the domain and range.

Domain: $(- \infty, - 2) \cup (- 2, 2) \cup (2, \infty), {x | x \neq 2, - 2}$

Range: $(- \infty, - \frac{3}{4}] \cup (0, \infty), {y | y \leq - \frac{3}{4}, y > 0}$

Step 6