Precalculus Examples

$f (x) = \sqrt{\frac{1}{3} x + 2}$

Step 1

Set the radicand in $\sqrt{\frac{1}{3} x + 2}$ greater than or equal to $0$ to find where the expression is defined.

$\frac{1}{3} x + 2 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Combine $\frac{1}{3}$ and $x$ .

$\frac{x}{3} + 2 \geq 0$

Step 2.2

Subtract $2$ from both sides of the inequality.

$\frac{x}{3} \geq - 2$

Step 2.3

Multiply both sides by $3$ .

$\frac{x}{3} \cdot 3 \geq - 2 \cdot 3$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Cancel the common factor of $3$ .

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Step 2.4.1.1.1

Cancel the common factor.

$\frac{x}{3} \cdot 3 \geq - 2 \cdot 3$

Step 2.4.1.1.2

Rewrite the expression.

$x \geq - 2 \cdot 3$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Multiply $- 2$ by $3$ .

$x \geq - 6$

Step 3

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

Interval Notation:

$[- 6, \infty)$

Set-Builder Notation:

${x | x \geq - 6}$

Step 4