Calculus Examples

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

Step 1

Find the domain to determine if the expression is continuous.

Tap for more steps...

Step 1.1

Set the radicand in $\sqrt{\frac{x}{x - 2}}$ greater than or equal to $0$ to find where the expression is defined.

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

Step 1.2

Solve for $x$ .

Tap for more steps...

Step 1.2.1

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x = 0$

$x - 2 = 0$

Step 1.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.3

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 0$

$x = 2$

Step 1.2.4

Consolidate the solutions.

$x = 0, 2$

Step 1.2.5

Find the domain of $\frac{x}{x - 2}$ .

Tap for more steps...

Step 1.2.5.1

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

$x - 2 = 0$

Step 1.2.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.5.3

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

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

Step 1.2.6

Use each root to create test intervals.

$x < 0$

$0 < x < 2$

$x > 2$

Step 1.2.7

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Step 1.2.7.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

Tap for more steps...

Step 1.2.7.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 1.2.7.1.2

Replace $x$ with $- 2$ in the original inequality.

$\frac{- 2}{(- 2) - 2} \geq 0$

Step 1.2.7.1.3

The left side $0.5$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 1.2.7.2

Test a value on the interval $0 < x < 2$ to see if it makes the inequality true.

Tap for more steps...

Step 1.2.7.2.1

Choose a value on the interval $0 < x < 2$ and see if this value makes the original inequality true.

$x = 1$

Step 1.2.7.2.2

Replace $x$ with $1$ in the original inequality.

$\frac{1}{(1) - 2} \geq 0$

Step 1.2.7.2.3

The left side $- 1$ is less than the right side $0$ , which means that the given statement is false.

False

Step 1.2.7.3

Test a value on the interval $x > 2$ to see if it makes the inequality true.

Tap for more steps...

Step 1.2.7.3.1

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

Step 1.2.7.3.2

Replace $x$ with $4$ in the original inequality.

$\frac{4}{(4) - 2} \geq 0$

Step 1.2.7.3.3

The left side $2$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 1.2.7.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < 0$ True

$0 < x < 2$ False

$x > 2$ True

$x < 0$ True

$0 < x < 2$ False

$x > 2$ True

Step 1.2.8

The solution consists of all of the true intervals.

$x \leq 0$ or $x > 2$

Step 1.3

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

$x - 2 = 0$

Step 1.4

Add $2$ to both sides of the equation.

$x = 2$

Step 1.5

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

Interval Notation:

$(- \infty, 0] \cup (2, \infty)$

Set-Builder Notation:

${x | x \leq 0, x > 2}$

Interval Notation:

$(- \infty, 0] \cup (2, \infty)$

Set-Builder Notation:

${x | x \leq 0, x > 2}$

Step 2

Since the domain is not all real numbers, $\sqrt{\frac{x}{x - 2}}$ is not continuous over all real numbers.

Not continuous

Step 3