Find the Domain x^2+y^2=81

$x^{2} + y^{2} = 81$

Step 1

Subtract $x^{2}$ from both sides of the equation.

$y^{2} = 81 - x^{2}$

Step 2

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

$y = \pm \sqrt{81 - x^{2}}$

Step 3

Simplify $\pm \sqrt{81 - x^{2}}$ .

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

$y = \pm \sqrt{9^{2} - x^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 9$ and $b = x$ .

$y = \pm \sqrt{(9 + x) (9 - x)}$

Step 4

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.

$y = \sqrt{(9 + x) (9 - x)}$

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

$y = - \sqrt{(9 + x) (9 - x)}$

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

$y = \sqrt{(9 + x) (9 - x)}$

$y = - \sqrt{(9 + x) (9 - x)}$

$y = \sqrt{(9 + x) (9 - x)}$

$y = - \sqrt{(9 + x) (9 - x)}$

Step 5

Set the radicand in $\sqrt{(9 + x) (9 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$(9 + x) (9 - x) \geq 0$

Step 6

Solve for $x$ .

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

$9 + x = 0$

$9 - x = 0$

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

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

$9 + x = 0$

Subtract $9$ from both sides of the equation.

$x = - 9$

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

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

$9 - x = 0$

Solve $9 - x = 0$ for $x$ .

Subtract $9$ from both sides of the equation.

$- x = - 9$

Divide each term in $- x = - 9$ by $- 1$ and simplify.

Divide each term in $- x = - 9$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 9}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 9}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 9}{- 1}$

Simplify the right side.

Divide $- 9$ by $- 1$ .

$x = 9$

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

$x = - 9, 9$

Use each root to create test intervals.

$x < - 9$

$- 9 < x < 9$

$x > 9$

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $x < - 9$ to see if it makes the inequality true.

Choose a value on the interval $x < - 9$ and see if this value makes the original inequality true.

$x = - 12$

Replace $x$ with $- 12$ in the original inequality.

$(9 - 12) (9 - (- 12)) \geq 0$

The left side $- 63$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 9 < x < 9$ to see if it makes the inequality true.

Choose a value on the interval $- 9 < x < 9$ and see if this value makes the original inequality true.

$x = 0$

Replace $x$ with $0$ in the original inequality.

$(9 + 0) (9 - (0)) \geq 0$

The left side $81$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 9$ to see if it makes the inequality true.

Choose a value on the interval $x > 9$ and see if this value makes the original inequality true.

$x = 12$

Replace $x$ with $12$ in the original inequality.

$(9 + 12) (9 - (12)) \geq 0$

The left side $- 63$ is less than the right side $0$ , which means that the given statement is false.

False

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 9$ False

$- 9 < x < 9$ True

$x > 9$ False

$x < - 9$ False

$- 9 < x < 9$ True

$x > 9$ False

The solution consists of all of the true intervals.

$- 9 \leq x \leq 9$

Step 7

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

Interval Notation:

$[- 9, 9]$

Set-Builder Notation:

${x | - 9 \leq x \leq 9}$

Step 8

Find the Domain x^2+y^2=81

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