Find the Domain and Range 1/( square root of 9-x^2)

$\frac{1}{\sqrt{9 - x^{2}}}$

Step 1

Set the radicand in $\sqrt{9 - x^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$9 - x^{2} \geq 0$

Step 2

Solve for $x$ .

Subtract $9$ from both sides of the inequality.

$- x^{2} \geq - 9$

Divide each term in $- x^{2} \geq - 9$ by $- 1$ and simplify.

Divide each term in $- x^{2} \geq - 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 9}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 9}{- 1}$

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

$x^{2} \leq \frac{- 9}{- 1}$

Simplify the right side.

Divide $- 9$ by $- 1$ .

$x^{2} \leq 9$

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

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

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

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

Simplify the right side.

Simplify $\sqrt{9}$ .

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

$| x | \leq \sqrt{3^{2}}$

Pull terms out from under the radical.

$| x | \leq | 3 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$| x | \leq 3$

Write $| x | \leq 3$ 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.

$x \leq 3$

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 $- 1$ .

$- x \leq 3$

Write as a piecewise.

${\begin{cases} x \leq 3 & x \geq 0 \\ - x \leq 3 & x < 0 \end{cases}$

Find the intersection of $x \leq 3$ and $x \geq 0$ .

$0 \leq x \leq 3$

Solve $- x \leq 3$ when $x < 0$ .

Divide each term in $- x \leq 3$ by $- 1$ and simplify.

Divide each term in $- x \leq 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{3}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{3}{- 1}$

Divide $x$ by $1$ .

$x \geq \frac{3}{- 1}$

Simplify the right side.

Divide $3$ by $- 1$ .

$x \geq - 3$

Find the intersection of $x \geq - 3$ and $x < 0$ .

$- 3 \leq x < 0$

Find the union of the solutions.

$- 3 \leq x \leq 3$

Step 3

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

$\sqrt{9 - x^{2}} = 0$

Step 4

Solve for $x$ .

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{9 - x^{2}}}^{2} = 0^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 - x^{2}}$ as ${(9 - x^{2})}^{\frac{1}{2}}$ .

${({(9 - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Simplify the left side.

Simplify ${({(9 - x^{2})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(9 - x^{2})}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(9 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(9 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Rewrite the expression.

${(9 - x^{2})}^{1} = 0^{2}$

Simplify.

$9 - x^{2} = 0^{2}$

Simplify the right side.

Raising $0$ to any positive power yields $0$ .

$9 - x^{2} = 0$

Solve for $x$ .

Subtract $9$ from both sides of the equation.

$- x^{2} = - 9$

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

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

$\frac{- x^{2}}{- 1} = \frac{- 9}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 9}{- 1}$

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

$x^{2} = \frac{- 9}{- 1}$

Simplify the right side.

Divide $- 9$ by $- 1$ .

$x^{2} = 9$

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

$x = \pm \sqrt{9}$

Simplify $\pm \sqrt{9}$ .

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

$x = \pm \sqrt{3^{2}}$

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

$x = \pm 3$

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.

$x = 3$

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

$x = - 3$

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

$x = 3, - 3$

Step 5

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

Interval Notation:

$(- 3, 3)$

Set-Builder Notation:

${x | - 3 < x < 3}$

Step 6

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

Interval Notation:

$[\frac{1}{3}, \infty)$

Set-Builder Notation:

${y | y \geq \frac{1}{3}}$

Step 7

Determine the domain and range.

Domain: $(- 3, 3), {x | - 3 < x < 3}$

Range: $[\frac{1}{3}, \infty), {y | y \geq \frac{1}{3}}$

Step 8

Find the Domain and Range 1/( square root of 9-x^2)

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