Precalculus Examples

$f (x) = \frac{\sqrt{x + 4}}{x^{2} + 5 x}$

Step 1

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

$x + 4 \geq 0$

Step 2

Subtract $4$ from both sides of the inequality.

$x \geq - 4$

Step 3

Set the denominator in $\frac{\sqrt{x + 4}}{x^{2} + 5 x}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 5 x = 0$

Step 4

Solve for $x$ .

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Step 4.1

Factor $x$ out of $x^{2} + 5 x$ .

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Step 4.1.1

Factor $x$ out of $x^{2}$ .

$x \cdot x + 5 x = 0$

Step 4.1.2

Factor $x$ out of $5 x$ .

$x \cdot x + x \cdot 5 = 0$

Step 4.1.3

Factor $x$ out of $x \cdot x + x \cdot 5$ .

$x (x + 5) = 0$

Step 4.2

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

$x = 0$

$x + 5 = 0$

Step 4.3

Set $x$ equal to $0$ .

$x = 0$

Step 4.4

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 4.4.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 4.4.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 4.5

The final solution is all the values that make $x (x + 5) = 0$ true.

$x = 0, - 5$

Step 5

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

Interval Notation:

$[- 4, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \geq - 4, x \neq 0}$

Step 6