Find the Domain and Range x^2+(y-1)^2=64

$x^{2} + {(y - 1)}^{2} = 64$

Step 1

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

${(y - 1)}^{2} = 64 - x^{2}$

Step 2

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

$y - 1 = \pm \sqrt{64 - x^{2}}$

Step 3

Simplify $\pm \sqrt{64 - x^{2}}$ .

Rewrite $64$ as $8^{2}$ .

$y - 1 = \pm \sqrt{8^{2} - x^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8$ and $b = x$ .

$y - 1 = \pm \sqrt{(8 + x) (8 - x)}$

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.

$y - 1 = \sqrt{(8 + x) (8 - x)}$

Add $1$ to both sides of the equation.

$y = \sqrt{(8 + x) (8 - x)} + 1$

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

$y - 1 = - \sqrt{(8 + x) (8 - x)}$

Add $1$ to both sides of the equation.

$y = - \sqrt{(8 + x) (8 - x)} + 1$

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

$y = \sqrt{(8 + x) (8 - x)} + 1$

$y = - \sqrt{(8 + x) (8 - x)} + 1$

$y = \sqrt{(8 + x) (8 - x)} + 1$

$y = - \sqrt{(8 + x) (8 - x)} + 1$

Step 5

Set the radicand in $\sqrt{(8 + x) (8 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$(8 + x) (8 - x) \geq 0$

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 $0$ .

$8 + x = 0$

$8 - x = 0$

Set $8 + x$ equal to $0$ and solve for $x$ .

Set $8 + x$ equal to $0$ .

$8 + x = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

Set $8 - x$ equal to $0$ and solve for $x$ .

Set $8 - x$ equal to $0$ .

$8 - x = 0$

Solve $8 - x = 0$ for $x$ .

Subtract $8$ from both sides of the equation.

$- x = - 8$

Divide each term in $- x = - 8$ by $- 1$ and simplify.

Divide each term in $- x = - 8$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 8}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 8}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 8}{- 1}$

Simplify the right side.

Divide $- 8$ by $- 1$ .

$x = 8$

The final solution is all the values that make $(8 + x) (8 - x) \geq 0$ true.

$x = - 8, 8$

Use each root to create test intervals.

$x < - 8$

$- 8 < x < 8$

$x > 8$

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 $x < - 8$ to see if it makes the inequality true.

Choose a value on the interval $x < - 8$ and see if this value makes the original inequality true.

$x = - 10$

Replace $x$ with $- 10$ in the original inequality.

$(8 - 10) (8 - (- 10)) \geq 0$

The left side $- 36$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $- 8 < x < 8$ to see if it makes the inequality true.

Choose a value on the interval $- 8 < x < 8$ and see if this value makes the original inequality true.

$x = 0$

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

$(8 + 0) (8 - (0)) \geq 0$

The left side $64$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 8$ to see if it makes the inequality true.

Choose a value on the interval $x > 8$ and see if this value makes the original inequality true.

$x = 10$

Replace $x$ with $10$ in the original inequality.

$(8 + 10) (8 - (10)) \geq 0$

The left side $- 36$ 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.

$x < - 8$ False

$- 8 < x < 8$ True

$x > 8$ False

$x < - 8$ False

$- 8 < x < 8$ True

$x > 8$ False

The solution consists of all of the true intervals.

$- 8 \leq x \leq 8$

Step 7

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

Interval Notation:

$[- 8, 8]$

Set-Builder Notation:

${x | - 8 \leq x \leq 8}$

Step 8

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[- 7, 9]$

Set-Builder Notation:

${y | - 7 \leq y \leq 9}$

Step 9

Determine the domain and range.

Domain: $[- 8, 8], {x | - 8 \leq x \leq 8}$

Range: $[- 7, 9], {y | - 7 \leq y \leq 9}$

Step 10

Find the Domain and Range x^2+(y-1)^2=64

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