Algebra Examples

$f (x) = \frac{1}{x^{2}}$ , $g (x) = \sqrt{2 + x}$

Step 1

Find the quotient of the functions.

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Step 1.1

Replace the function designators with the actual functions in $\frac{f (x)}{g (x)}$ .

$\frac{\frac{1}{x^{2}}}{\sqrt{2 + x}}$

Step 1.2

Simplify.

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Step 1.2.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{x^{2}} \cdot \frac{1}{\sqrt{2 + x}}$

Step 1.2.2

Combine.

$\frac{1 \cdot 1}{x^{2} \sqrt{2 + x}}$

Step 1.2.3

Multiply $1$ by $1$ .

$\frac{1}{x^{2} \sqrt{2 + x}}$

Step 1.2.4

Multiply $\frac{1}{x^{2} \sqrt{2 + x}}$ by $\frac{\sqrt{2 + x}}{\sqrt{2 + x}}$ .

$\frac{1}{x^{2} \sqrt{2 + x}} \cdot \frac{\sqrt{2 + x}}{\sqrt{2 + x}}$

Step 1.2.5

Combine and simplify the denominator.

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Step 1.2.5.1

Multiply $\frac{1}{x^{2} \sqrt{2 + x}}$ by $\frac{\sqrt{2 + x}}{\sqrt{2 + x}}$ .

$\frac{\sqrt{2 + x}}{x^{2} \sqrt{2 + x} \sqrt{2 + x}}$

Step 1.2.5.2

Move $\sqrt{2 + x}$ .

$\frac{\sqrt{2 + x}}{x^{2} (\sqrt{2 + x} \sqrt{2 + x})}$

Step 1.2.5.3

Raise $\sqrt{2 + x}$ to the power of $1$ .

$\frac{\sqrt{2 + x}}{x^{2} ({\sqrt{2 + x}}^{1} \sqrt{2 + x})}$

Step 1.2.5.4

Raise $\sqrt{2 + x}$ to the power of $1$ .

$\frac{\sqrt{2 + x}}{x^{2} ({\sqrt{2 + x}}^{1} {\sqrt{2 + x}}^{1})}$

Step 1.2.5.5

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

$\frac{\sqrt{2 + x}}{x^{2} {\sqrt{2 + x}}^{1 + 1}}$

Step 1.2.5.6

Add $1$ and $1$ .

$\frac{\sqrt{2 + x}}{x^{2} {\sqrt{2 + x}}^{2}}$

Step 1.2.5.7

Rewrite ${\sqrt{2 + x}}^{2}$ as $2 + x$ .

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Step 1.2.5.7.1

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

$\frac{\sqrt{2 + x}}{x^{2} {({(2 + x)}^{\frac{1}{2}})}^{2}}$

Step 1.2.5.7.2

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

$\frac{\sqrt{2 + x}}{x^{2} {(2 + x)}^{\frac{1}{2} \cdot 2}}$

Step 1.2.5.7.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{2 + x}}{x^{2} {(2 + x)}^{\frac{2}{2}}}$

Step 1.2.5.7.4

Cancel the common factor of $2$ .

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Step 1.2.5.7.4.1

Cancel the common factor.

$\frac{\sqrt{2 + x}}{x^{2} {(2 + x)}^{\frac{2}{2}}}$

Step 1.2.5.7.4.2

Rewrite the expression.

$\frac{\sqrt{2 + x}}{x^{2} {(2 + x)}^{1}}$

Step 1.2.5.7.5

Simplify.

$\frac{\sqrt{2 + x}}{x^{2} (2 + x)}$

Step 2

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

$2 + x \geq 0$

Step 3

Subtract $2$ from both sides of the inequality.

$x \geq - 2$

Step 4

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

$x^{2} (2 + x) = 0$

Step 5

Solve for $x$ .

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Step 5.1

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

$x^{2} = 0$

$2 + x = 0$

Step 5.2

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 5.2.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 5.2.2

Solve $x^{2} = 0$ for $x$ .

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Step 5.2.2.1

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

$x = \pm \sqrt{0}$

Step 5.2.2.2

Simplify $\pm \sqrt{0}$ .

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Step 5.2.2.2.1

Rewrite $0$ as $0^{2}$ .

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

Step 5.2.2.2.2

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

$x = \pm 0$

Step 5.2.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5.3

Set $2 + x$ equal to $0$ and solve for $x$ .

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Step 5.3.1

Set $2 + x$ equal to $0$ .

$2 + x = 0$

Step 5.3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5.4

The final solution is all the values that make $x^{2} (2 + x) = 0$ true.

$x = 0, - 2$

Step 6

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

Interval Notation:

$(- 2, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x > - 2, x \neq 0}$

Step 7