Precalculus Examples

$h (z) = \frac{1}{\sqrt{z}}$

Step 1

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

$z \geq 0$

Step 2

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

$\sqrt{z} = 0$

Step 3

Solve for $z$ .

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Step 3.1

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

${\sqrt{z}}^{2} = 0^{2}$

Step 3.2

Simplify each side of the equation.

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Step 3.2.1

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

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

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

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Step 3.2.2.1.1.1

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

$z^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.2.1

Cancel the common factor.

$z^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 3.2.2.1.1.2.2

Rewrite the expression.

$z^{1} = 0^{2}$

Step 3.2.2.1.2

Simplify.

$z = 0^{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

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

$z = 0$

Step 4

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${z | z > 0}$

Step 5