Algebra Examples

$g (x) = \sqrt{\frac{1}{3} (x + 4)} - 1$

Step 1

Set the radicand in $\sqrt{\frac{1}{3} \cdot (x + 4)}$ greater than or equal to $0$ to find where the expression is defined.

$\frac{1}{3} \cdot (x + 4) \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Multiply each term in $\frac{1}{3} \cdot (x + 4) \geq 0$ by $3$ to eliminate the fractions.

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Step 2.1.1

Multiply each term in $\frac{1}{3} \cdot (x + 4) \geq 0$ by $3$ .

$\frac{1}{3} \cdot (x + 4) \cdot 3 \geq 0 \cdot 3$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Apply the distributive property.

$(\frac{1}{3} x + \frac{1}{3} \cdot 4) \cdot 3 \geq 0 \cdot 3$

Step 2.1.2.2

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

$(\frac{x}{3} + \frac{1}{3} \cdot 4) \cdot 3 \geq 0 \cdot 3$

Step 2.1.2.3

Combine $\frac{1}{3}$ and $4$ .

$(\frac{x}{3} + \frac{4}{3}) \cdot 3 \geq 0 \cdot 3$

Step 2.1.2.4

Apply the distributive property.

$\frac{x}{3} \cdot 3 + \frac{4}{3} \cdot 3 \geq 0 \cdot 3$

Step 2.1.2.5

Cancel the common factor of $3$ .

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Step 2.1.2.5.1

Cancel the common factor.

$\frac{x}{3} \cdot 3 + \frac{4}{3} \cdot 3 \geq 0 \cdot 3$

Step 2.1.2.5.2

Rewrite the expression.

$x + \frac{4}{3} \cdot 3 \geq 0 \cdot 3$

Step 2.1.2.6

Cancel the common factor of $3$ .

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Step 2.1.2.6.1

Cancel the common factor.

$x + \frac{4}{3} \cdot 3 \geq 0 \cdot 3$

Step 2.1.2.6.2

Rewrite the expression.

$x + 4 \geq 0 \cdot 3$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Multiply $0$ by $3$ .

$x + 4 \geq 0$

Step 2.2

Subtract $4$ from both sides of the inequality.

$x \geq - 4$

Step 3

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

Interval Notation:

$[- 4, \infty)$

Set-Builder Notation:

${x | x \geq - 4}$

Step 4

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

Interval Notation:

$[- 1, \infty)$

Set-Builder Notation:

${y | y \geq - 1}$

Step 5

Determine the domain and range.

Domain: $[- 4, \infty), {x | x \geq - 4}$

Range: $[- 1, \infty), {y | y \geq - 1}$

Step 6