Precalculus Examples

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

Step 1

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

$x - 9 \geq 0$

Step 2

Add $9$ to both sides of the inequality.

$x \geq 9$

Step 3

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

$\sqrt{x - 9} = 0$

Step 4

Solve for $x$ .

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Step 4.1

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

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

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

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

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

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

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Step 4.2.2.1.1

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

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Step 4.2.2.1.1.1

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

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

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

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

Step 4.2.2.1.1.2.2

Rewrite the expression.

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

Step 4.2.2.1.2

Simplify.

$x - 9 = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

$x - 9 = 0$

Step 4.3

Add $9$ to both sides of the equation.

$x = 9$

Step 5

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

Interval Notation:

$(9, \infty)$

Set-Builder Notation:

${x | x > 9}$

Step 6

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${y | y > 0}$

Step 7

Determine the domain and range.

Domain: $(9, \infty), {x | x > 9}$

Range: $(0, \infty), {y | y > 0}$

Step 8