Algebra Examples

$y = 2 \sqrt{3 x + 4} - 5$

Step 1

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

$3 x + 4 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $4$ from both sides of the inequality.

$3 x \geq - 4$

Step 2.2

Divide each term in $3 x \geq - 4$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 x \geq - 4$ by $3$ .

$\frac{3 x}{3} \geq \frac{- 4}{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{- 4}{3}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 4}{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$x \geq - \frac{4}{3}$

Step 3

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

Interval Notation:

$[- \frac{4}{3}, \infty)$

Set-Builder Notation:

${x | x \geq - \frac{4}{3}}$

Step 4

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

Interval Notation:

$[- 5, \infty)$

Set-Builder Notation:

${y | y \geq - 5}$

Step 5

Determine the domain and range.

Domain: $[- \frac{4}{3}, \infty), {x | x \geq - \frac{4}{3}}$

Range: $[- 5, \infty), {y | y \geq - 5}$

Step 6