Find the Domain and Range f(x)=7+ square root of 3x-6

$f (x) = 7 + \sqrt{3 x - 6}$

Set the radicand in $\sqrt{3 x - 6}$ greater than or equal to $0$ to find where the expression is defined.

$3 x - 6 \geq 0$

Solve for $x$ .

Add $6$ to both sides of the inequality.

$3 x \geq 6$

Divide each term in $3 x \geq 6$ by $3$ and simplify.

Divide each term in $3 x \geq 6$ by $3$ .

$\frac{3 x}{3} \geq \frac{6}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{6}{3}$

Divide $x$ by $1$ .

$x \geq \frac{6}{3}$

Simplify the right side.

Divide $6$ by $3$ .

$x \geq 2$

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

Interval Notation:

$[2, \infty)$

Set-Builder Notation:

${x | x \geq 2}$

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

Interval Notation:

$[7, \infty)$

Set-Builder Notation:

${y | y \geq 7}$

Determine the domain and range.

Domain: $[2, \infty), {x | x \geq 2}$

Range: $[7, \infty), {y | y \geq 7}$