Precalculus Examples

$f (x) = \sqrt{9 - \sqrt{- x}}$

Step 1

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

$- x \geq 0$

Step 2

Divide each term in $- x \geq 0$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- x \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{0}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{0}{- 1}$

Step 2.2.2

Divide $x$ by $1$ .

$x \leq \frac{0}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $- 1$ .

$x \leq 0$

Step 3

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

$9 - \sqrt{- x} \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $9$ from both sides of the inequality.

$- \sqrt{- x} \geq - 9$

Step 4.2

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

${(- \sqrt{- x})}^{2} \geq {(- 9)}^{2}$

Step 4.3

Simplify each side of the inequality.

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Step 4.3.1

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

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

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

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Step 4.3.2.1.1

Apply the product rule to $- x$ .

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

Step 4.3.2.1.2

Multiply $- 1$ by ${(- 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.3.2.1.2.1

Move ${(- 1)}^{\frac{1}{2}}$ .

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

Step 4.3.2.1.2.2

Multiply ${(- 1)}^{\frac{1}{2}}$ by $- 1$ .

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Step 4.3.2.1.2.2.1

Raise $- 1$ to the power of $1$ .

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

Step 4.3.2.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${({(- 1)}^{\frac{1}{2} + 1} x^{\frac{1}{2}})}^{2} \geq {(- 9)}^{2}$

Step 4.3.2.1.2.3

Write $1$ as a fraction with a common denominator.

${({(- 1)}^{\frac{1}{2} + \frac{2}{2}} x^{\frac{1}{2}})}^{2} \geq {(- 9)}^{2}$

Step 4.3.2.1.2.4

Combine the numerators over the common denominator.

${({(- 1)}^{\frac{1 + 2}{2}} x^{\frac{1}{2}})}^{2} \geq {(- 9)}^{2}$

Step 4.3.2.1.2.5

Add $1$ and $2$ .

${({(- 1)}^{\frac{3}{2}} x^{\frac{1}{2}})}^{2} \geq {(- 9)}^{2}$

Step 4.3.2.1.3

Rewrite ${(- 1)}^{\frac{3}{2}}$ as $\sqrt{{(- 1)}^{3}}$ .

${(\sqrt{{(- 1)}^{3}} x^{\frac{1}{2}})}^{2} \geq {(- 9)}^{2}$

Step 4.3.2.1.4

Raise $- 1$ to the power of $3$ .

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

Step 4.3.2.1.5

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

${(i x^{\frac{1}{2}})}^{2} \geq {(- 9)}^{2}$

Step 4.3.2.1.6

Apply the product rule to $i x^{\frac{1}{2}}$ .

$i^{2} {(x^{\frac{1}{2}})}^{2} \geq {(- 9)}^{2}$

Step 4.3.2.1.7

Rewrite $i^{2}$ as $- 1$ .

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

Step 4.3.2.1.8

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

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Step 4.3.2.1.8.1

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

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

Step 4.3.2.1.8.2

Cancel the common factor of $2$ .

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Step 4.3.2.1.8.2.1

Cancel the common factor.

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

Step 4.3.2.1.8.2.2

Rewrite the expression.

$- x^{1} \geq {(- 9)}^{2}$

Step 4.3.2.1.9

Simplify.

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

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Raise $- 9$ to the power of $2$ .

$- x \geq 81$

Step 4.4

Divide each term in $- x \geq 81$ by $- 1$ and simplify.

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Step 4.4.1

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

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

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Dividing two negative values results in a positive value.

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

Step 4.4.2.2

Divide $x$ by $1$ .

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

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Divide $81$ by $- 1$ .

$x \leq - 81$

Step 4.5

Find the domain of $9 - \sqrt{- x}$ .

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Step 4.5.1

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

$- x \geq 0$

Step 4.5.2

Divide each term in $- x \geq 0$ by $- 1$ and simplify.

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Step 4.5.2.1

Divide each term in $- x \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{0}{- 1}$

Step 4.5.2.2

Simplify the left side.

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Step 4.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{0}{- 1}$

Step 4.5.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{0}{- 1}$

Step 4.5.2.3

Simplify the right side.

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Step 4.5.2.3.1

Divide $0$ by $- 1$ .

$x \leq 0$

Step 4.5.3

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

$(- \infty, 0]$

Step 4.6

Use each root to create test intervals.

$x < - 81$

$- 81 < x < 0$

$x > 0$

Step 4.7

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

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Step 4.7.1

Test a value on the interval $x < - 81$ to see if it makes the inequality true.

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Step 4.7.1.1

Choose a value on the interval $x < - 81$ and see if this value makes the original inequality true.

$x = - 84$

Step 4.7.1.2

Replace $x$ with $- 84$ in the original inequality.

$9 - \sqrt{- (- 84)} \geq 0$

Step 4.7.1.3

The left side $- 0.16515138$ is less than the right side $0$ , which means that the given statement is false.

False

Step 4.7.2

Test a value on the interval $- 81 < x < 0$ to see if it makes the inequality true.

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Step 4.7.2.1

Choose a value on the interval $- 81 < x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 4.7.2.2

Replace $x$ with $- 2$ in the original inequality.

$9 - \sqrt{- (- 2)} \geq 0$

Step 4.7.2.3

The left side $7.58578643$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 4.7.3

Test a value on the interval $x > 0$ to see if it makes the inequality true.

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Step 4.7.3.1

Choose a value on the interval $x > 0$ and see if this value makes the original inequality true.

$x = 2$

Step 4.7.3.2

Replace $x$ with $2$ in the original inequality.

$9 - \sqrt{- (2)} \geq 0$

Step 4.7.3.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 4.7.4

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

$x < - 81$ False

$- 81 < x < 0$ True

$x > 0$ False

$x < - 81$ False

$- 81 < x < 0$ True

$x > 0$ False

Step 4.8

The solution consists of all of the true intervals.

$- 81 \leq x \leq 0$

Step 5

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

Interval Notation:

$[- 81, 0]$

Set-Builder Notation:

${x | - 81 \leq x \leq 0}$

Step 6