Precalculus Examples

$f (x) = \frac{\sqrt{x + 6}}{(x + 7) (x - 9)}$

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 $\frac{\sqrt{x + 6}}{(x + 7) (x - 9)}$ equal to $0$ to find where the expression is undefined.

$(x + 7) (x - 9) = 0$

Step 4

Solve for $x$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 7 = 0$

$x - 9 = 0$

Step 4.2

Set $x + 7$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Step 4.2.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 4.3

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 4.3.2

Add $9$ to both sides of the equation.

$x = 9$

Step 4.4

The final solution is all the values that make $(x + 7) (x - 9) = 0$ true.

$x = - 7, 9$

Step 5

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

Interval Notation:

$[- 6, 9) \cup (9, \infty)$

Set-Builder Notation:

${x | x \geq - 6, x \neq 9}$

Step 6