Precalculus Examples

$\sqrt{\sqrt{x}}$

Step 1

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

$x \geq 0$

Step 2

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

$\sqrt{x} \geq 0$

Step 3

Solve for $x$ .

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Step 3.1

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

${\sqrt{x}}^{2} \geq 0^{2}$

Step 3.2

Simplify each side of the inequality.

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Step 3.2.1

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

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

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

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Step 3.2.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} \geq 0^{2}$

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.

$x^{\frac{1}{2} \cdot 2} \geq 0^{2}$

Step 3.2.2.1.1.2.2

Rewrite the expression.

$x^{1} \geq 0^{2}$

Step 3.2.2.1.2

Simplify.

$x \geq 0^{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

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

$x \geq 0$

Step 3.3

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

$[0, \infty)$

Step 3.4

The solution consists of all of the true intervals.

$x \geq 0$

Step 4

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 5

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${y | y \geq 0}$

Step 6

Determine the domain and range.

Domain: $[0, \infty), {x | x \geq 0}$

Range: $[0, \infty), {y | y \geq 0}$

Step 7