Algebra Examples

$g (x) = \sqrt{\frac{1}{3} (x - 1)}$

Step 1

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

$\frac{1}{3} \cdot (x - 1) \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Simplify $\frac{1}{3} \cdot (x - 1)$ .

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Step 2.1.1

Apply the distributive property.

$\frac{1}{3} x + \frac{1}{3} \cdot - 1 \geq 0$

Step 2.1.2

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

$\frac{x}{3} + \frac{1}{3} \cdot - 1 \geq 0$

Step 2.1.3

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

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

Step 2.1.4

Move the negative in front of the fraction.

$\frac{x}{3} - \frac{1}{3} \geq 0$

Step 2.2

Add $\frac{1}{3}$ to both sides of the inequality.

$\frac{x}{3} \geq \frac{1}{3}$

Step 2.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x \geq 1$

Step 3

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

Interval Notation:

$[1, \infty)$

Set-Builder Notation:

${x | x \geq 1}$

Step 4

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 5

Determine the domain and range.

Domain: $[1, \infty), {x | x \geq 1}$

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

Step 6