Precalculus Examples

$f (x) = \frac{\sqrt{x + 6}}{6 + x}$

Step 1

Set the radicand in $\sqrt{x + 6}$ greater than or equal to $0$ to find where the expression is defined.

$x + 6 \geq 0$

Step 2

Subtract $6$ from both sides of the inequality.

$x \geq - 6$

Step 3

Set the denominator in $\frac{\sqrt{x + 6}}{6 + x}$ equal to $0$ to find where the expression is undefined.

$6 + x = 0$

Step 4

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 5

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

Interval Notation:

$(- 6, \infty)$

Set-Builder Notation:

${x | x > - 6}$

Step 6

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${y | y > 0}$

Step 7

Determine the domain and range.

Domain: $(- 6, \infty), {x | x > - 6}$

Range: $(0, \infty), {y | y > 0}$

Step 8