Precalculus Examples

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

Step 1

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

$x - \frac{4}{x} \geq 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x, 1$

Step 2.1.2

The LCM of one and any expression is the expression.

$x$

Step 2.2

Multiply each term in $x - \frac{4}{x} \geq 0$ by $x$ to eliminate the fractions.

Tap for more steps...

Step 2.2.1

Multiply each term in $x - \frac{4}{x} \geq 0$ by $x$ .

$x \cdot x - \frac{4}{x} x \geq 0 x$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.2.1.1

Multiply $x$ by $x$ .

$x^{2} - \frac{4}{x} x \geq 0 x$

Step 2.2.2.1.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.2.2.1.2.1

Move the leading negative in $- \frac{4}{x}$ into the numerator.

$x^{2} + \frac{- 4}{x} x \geq 0 x$

Step 2.2.2.1.2.2

Cancel the common factor.

$x^{2} + \frac{- 4}{x} x \geq 0 x$

Step 2.2.2.1.2.3

Rewrite the expression.

$x^{2} - 4 \geq 0 x$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Multiply $0$ by $x$ .

$x^{2} - 4 \geq 0$

Step 2.3

Solve the inequality.

Tap for more steps...

Step 2.3.1

Add $4$ to both sides of the inequality.

$x^{2} \geq 4$

Step 2.3.2

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

$\sqrt{x^{2}} \geq \sqrt{4}$

Step 2.3.3

Simplify the equation.

Tap for more steps...

Step 2.3.3.1

Simplify the left side.

Tap for more steps...

Step 2.3.3.1.1

Pull terms out from under the radical.

$| x | \geq \sqrt{4}$

Step 2.3.3.2

Simplify the right side.

Tap for more steps...

Step 2.3.3.2.1

Simplify $\sqrt{4}$ .

Tap for more steps...

Step 2.3.3.2.1.1

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

$| x | \geq \sqrt{2^{2}}$

Step 2.3.3.2.1.2

Pull terms out from under the radical.

$| x | \geq | 2 |$

Step 2.3.3.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$| x | \geq 2$

Step 2.3.4

Write $| x | \geq 2$ as a piecewise.

Tap for more steps...

Step 2.3.4.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 2.3.4.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \geq 2$

Step 2.3.4.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 2.3.4.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \geq 2$

Step 2.3.4.5

Write as a piecewise.

${\begin{cases} x \geq 2 & x \geq 0 \\ - x \geq 2 & x < 0 \end{cases}$

Step 2.3.5

Find the intersection of $x \geq 2$ and $x \geq 0$ .

$x \geq 2$

Step 2.3.6

Divide each term in $- x \geq 2$ by $- 1$ and simplify.

Tap for more steps...

Step 2.3.6.1

Divide each term in $- x \geq 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 2.3.6.2

Simplify the left side.

Tap for more steps...

Step 2.3.6.2.1

Dividing two negative values results in a positive value.

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

Step 2.3.6.2.2

Divide $x$ by $1$ .

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

Step 2.3.6.3

Simplify the right side.

Tap for more steps...

Step 2.3.6.3.1

Divide $2$ by $- 1$ .

$x \leq - 2$

Step 2.3.7

Find the union of the solutions.

$x \leq - 2$ or $x \geq 2$

Step 3

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

$x = 0$

Step 4

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

Interval Notation:

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

Set-Builder Notation:

${x | x \leq - 2, x \geq 2}$

Step 5