Precalculus Examples

$f (x) = \frac{x}{(x - 3) \sqrt{x - 2}}$

Step 1

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

$x - 2 \geq 0$

Step 2

Add $2$ to both sides of the inequality.

$x \geq 2$

Step 3

Set the denominator in $\frac{x}{(x - 3) \sqrt{x - 2}}$ equal to $0$ to find where the expression is undefined.

$(x - 3) \sqrt{x - 2} = 0$

Step 4

Solve for $x$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 3 = 0$

$\sqrt{x - 2} = 0$

Step 4.2

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4.3

Set $\sqrt{x - 2}$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $\sqrt{x - 2}$ equal to $0$ .

$\sqrt{x - 2} = 0$

Step 4.3.2

Solve $\sqrt{x - 2} = 0$ for $x$ .

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Step 4.3.2.1

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

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

Step 4.3.2.2

Simplify each side of the equation.

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Step 4.3.2.2.1

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

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

Step 4.3.2.2.2

Simplify the left side.

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Step 4.3.2.2.2.1

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

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Step 4.3.2.2.2.1.1

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

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Step 4.3.2.2.2.1.1.1

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

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

Step 4.3.2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.3.2.2.2.1.1.2.1

Cancel the common factor.

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

Step 4.3.2.2.2.1.1.2.2

Rewrite the expression.

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

Step 4.3.2.2.2.1.2

Simplify.

$x - 2 = 0^{2}$

Step 4.3.2.2.3

Simplify the right side.

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Step 4.3.2.2.3.1

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

$x - 2 = 0$

Step 4.3.2.3

Add $2$ to both sides of the equation.

$x = 2$

Step 4.4

The final solution is all the values that make $(x - 3) \sqrt{x - 2} = 0$ true.

$x = 3, 2$

Step 5

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

Interval Notation:

$(2, 3) \cup (3, \infty)$

Set-Builder Notation:

${x | x > 2, x \neq 3}$

Step 6