Find the Domain and Range y=- square root of 81-x^2

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

Step 1

Set the radicand in $\sqrt{81 - x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$81 - x^{2} \geq 0$

Step 2

Solve for $x$ .

Subtract $81$ from both sides of the inequality.

$- x^{2} \geq - 81$

Divide each term in $- x^{2} \geq - 81$ by $- 1$ and simplify.

Divide each term in $- x^{2} \geq - 81$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 81}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 81}{- 1}$

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

$x^{2} \leq \frac{- 81}{- 1}$

Simplify the right side.

Divide $- 81$ by $- 1$ .

$x^{2} \leq 81$

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

$\sqrt{x^{2}} \leq \sqrt{81}$

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

$| x | \leq \sqrt{81}$

Simplify the right side.

Simplify $\sqrt{81}$ .

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

$| x | \leq \sqrt{9^{2}}$

Pull terms out from under the radical.

$| x | \leq | 9 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $9$ is $9$ .

$| x | \leq 9$

Write $| x | \leq 9$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 9$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq 9$

Write as a piecewise.

${\begin{cases} x \leq 9 & x \geq 0 \\ - x \leq 9 & x < 0 \end{cases}$

Find the intersection of $x \leq 9$ and $x \geq 0$ .

$0 \leq x \leq 9$

Solve $- x \leq 9$ when $x < 0$ .

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

Divide each term in $- x \leq 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $9$ by $- 1$ .

$x \geq - 9$

Find the intersection of $x \geq - 9$ and $x < 0$ .

$- 9 \leq x < 0$

Find the union of the solutions.

$- 9 \leq x \leq 9$

Step 3

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 4

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

Interval Notation:

$[- 9, 0]$

Set-Builder Notation:

${y | - 9 \leq y \leq 0}$

Step 5

Determine the domain and range.

Domain: $[- 9, 9], {x | - 9 \leq x \leq 9}$

Range: $[- 9, 0], {y | - 9 \leq y \leq 0}$

Step 6

Find the Domain and Range y=- square root of 81-x^2

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