Precalculus Examples

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

Step 1

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

$x^{2} \geq 0$

Step 2

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 3

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

$\sqrt{x^{2}} - a^{2} = 0$

Step 4

Solve for $b$ .

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Step 4.1

Pull terms out from under the radical, assuming positive real numbers.

$x - a^{2} = 0$

Step 4.2

Add $a^{2}$ to both sides of the equation.

$x = a^{2}$

Step 5

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${b | b \in ℝ}$