Precalculus Examples

$\frac{1}{\sqrt{x + 10}}$

Step 1

Set the radicand in $\sqrt{x + 10}$ greater than or equal to $0$ to find where the expression is defined.

$x + 10 \geq 0$

Step 2

Subtract $10$ from both sides of the inequality.

$x \geq - 10$

Step 3

Set the denominator in $\frac{1}{\sqrt{x + 10}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x + 10} = 0$

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.

${\sqrt{x + 10}}^{2} = 0^{2}$

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 $\sqrt{x + 10}$ as ${(x + 10)}^{\frac{1}{2}}$ .

${({(x + 10)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${({(x + 10)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1

Multiply the exponents in ${({(x + 10)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1.1

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

${(x + 10)}^{\frac{1}{2} \cdot 2} = 0^{2}$

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.

${(x + 10)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(x + 10)}^{1} = 0^{2}$

Step 4.2.2.1.2

Simplify.

$x + 10 = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

$x + 10 = 0$

Step 4.3

Subtract $10$ from both sides of the equation.

$x = - 10$

Step 5

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

Interval Notation:

$(- 10, \infty)$

Set-Builder Notation:

${x | x > - 10}$

Step 6

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

Interval Notation:

$(- \infty, 0)$

Set-Builder Notation:

${y | y < 0}$

Step 7

Determine the domain and range.

Domain: $(- 10, \infty), {x | x > - 10}$

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

Step 8