Trigonometry Examples

$f (x) = \frac{4 x - 6}{x^{2} - 5}$

Step 1

Set the denominator in $\frac{4 x - 6}{x^{2} - 5}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 5 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $5$ to both sides of the equation.

$x^{2} = 5$

Step 2.2

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

$x = \pm \sqrt{5}$

Step 2.3

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

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Step 2.3.1

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

$x = \sqrt{5}$

Step 2.3.2

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

$x = - \sqrt{5}$

Step 2.3.3

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

$x = \sqrt{5}, - \sqrt{5}$

Step 3

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

Interval Notation:

$(- \infty, - \sqrt{5}) \cup (- \sqrt{5}, \sqrt{5}) \cup (\sqrt{5}, \infty)$

Set-Builder Notation:

${x | x \neq \sqrt{5}, - \sqrt{5}}$

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: $(- \infty, - \sqrt{5}) \cup (- \sqrt{5}, \sqrt{5}) \cup (\sqrt{5}, \infty), {x | x \neq \sqrt{5}, - \sqrt{5}}$

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

Step 6