Trigonometry Examples

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

Step 1

Set the radicand in $\sqrt{4 x - 3}$ greater than or equal to $0$ to find where the expression is defined.

$4 x - 3 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $3$ to both sides of the inequality.

$4 x \geq 3$

Step 2.2

Divide each term in $4 x \geq 3$ by $4$ and simplify.

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Step 2.2.1

Divide each term in $4 x \geq 3$ by $4$ .

$\frac{4 x}{4} \geq \frac{3}{4}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{3}{4}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{3}{4}$

Step 3

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

$x^{2} - 4 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $4$ to both sides of the equation.

$x^{2} = 4$

Step 4.2

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

$x = \pm \sqrt{4}$

Step 4.3

Simplify $\pm \sqrt{4}$ .

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Step 4.3.1

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

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

Step 4.3.2

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

$x = \pm 2$

Step 4.4

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

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Step 4.4.1

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

$x = 2$

Step 4.4.2

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

$x = - 2$

Step 4.4.3

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

$x = 2, - 2$

Step 5

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

Interval Notation:

$[\frac{3}{4}, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \geq \frac{3}{4}, x \neq 2}$

Step 6