Find the Domain and Range y = log base 3 of 2x-6+3

$y = \log_{3} (2 x - 6) + 3$

Step 1

Set the argument in $\log_{3} (2 x - 6)$ greater than $0$ to find where the expression is defined.

$2 x - 6 > 0$

Step 2

Solve for $x$ .

Add $6$ to both sides of the inequality.

$2 x > 6$

Divide each term in $2 x > 6$ by $2$ and simplify.

Divide each term in $2 x > 6$ by $2$ .

$\frac{2 x}{2} > \frac{6}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} > \frac{6}{2}$

Divide $x$ by $1$ .

$x > \frac{6}{2}$

Simplify the right side.

Divide $6$ by $2$ .

$x > 3$

Step 3

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

Interval Notation:

$(3, \infty)$

Set-Builder Notation:

${x | x > 3}$

Step 4

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

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Step 5

Determine the domain and range.

Domain: $(3, \infty), {x | x > 3}$

Range: $(- \infty, \infty), {y | y \in ℝ}$

Step 6