Find the Domain f(x)=x/(x^2+81)

$f (x) = \frac{x}{x^{2} + 81}$

Step 1

Set the denominator in $\frac{x}{x^{2} + 81}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 81 = 0$

Step 2

Solve for $x$ .

Subtract $81$ from both sides of the equation.

$x^{2} = - 81$

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

$x = \pm \sqrt{- 81}$

Simplify $\pm \sqrt{- 81}$ .

Rewrite $- 81$ as $- 1 (81)$ .

$x = \pm \sqrt{- 1 (81)}$

Rewrite $\sqrt{- 1 (81)}$ as $\sqrt{- 1} \cdot \sqrt{81}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{81}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{81}$

Rewrite $81$ as $9^{2}$ .

$x = \pm i \cdot \sqrt{9^{2}}$

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

$x = \pm i \cdot 9$

Move $9$ to the left of $i$ .

$x = \pm 9 i$

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.

$x = 9 i$

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

$x = - 9 i$

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

$x = 9 i, - 9 i$

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4