Find the Domain (4w-3)/(9w^2-1)

$\frac{4 w - 3}{9 w^{2} - 1}$

Step 1

Set the denominator in $\frac{4 w - 3}{9 w^{2} - 1}$ equal to $0$ to find where the expression is undefined.

$9 w^{2} - 1 = 0$

Step 2

Solve for $w$ .

Add $1$ to both sides of the equation.

$9 w^{2} = 1$

Divide each term in $9 w^{2} = 1$ by $9$ and simplify.

Divide each term in $9 w^{2} = 1$ by $9$ .

$\frac{9 w^{2}}{9} = \frac{1}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 w^{2}}{9} = \frac{1}{9}$

Divide $w^{2}$ by $1$ .

$w^{2} = \frac{1}{9}$

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$w = \pm \sqrt{\frac{1}{9}}$

Simplify $\pm \sqrt{\frac{1}{9}}$ .

Rewrite $\sqrt{\frac{1}{9}}$ as $\frac{\sqrt{1}}{\sqrt{9}}$ .

$w = \pm \frac{\sqrt{1}}{\sqrt{9}}$

Any root of $1$ is $1$ .

$w = \pm \frac{1}{\sqrt{9}}$

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$w = \pm \frac{1}{\sqrt{3^{2}}}$

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

$w = \pm \frac{1}{3}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$w = \frac{1}{3}$

Next, use the negative value of the $\pm$ to find the second solution.

$w = - \frac{1}{3}$

The complete solution is the result of both the positive and negative portions of the solution.

$w = \frac{1}{3}, - \frac{1}{3}$

Step 3

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

Interval Notation:

$(- \infty, - \frac{1}{3}) \cup (- \frac{1}{3}, \frac{1}{3}) \cup (\frac{1}{3}, \infty)$

Set-Builder Notation:

${w | w \neq \frac{1}{3}, - \frac{1}{3}}$

Step 4