Find the Domain and Range G(x) = log base 3 of 2x+1-2

$G (x) = \log_{3} (2 x + 1) - 2$

Step 1

Set the argument in $\log_{3} (2 x + 1)$ greater than $0$ to find where the expression is defined.

$2 x + 1 > 0$

Step 2

Solve for $x$ .

Subtract $1$ from both sides of the inequality.

$2 x > - 1$

Divide each term in $2 x > - 1$ by $2$ and simplify.

Divide each term in $2 x > - 1$ by $2$ .

$\frac{2 x}{2} > \frac{- 1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} > \frac{- 1}{2}$

Divide $x$ by $1$ .

$x > \frac{- 1}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x > - \frac{1}{2}$

Step 3

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

Interval Notation:

$(- \frac{1}{2}, \infty)$

Set-Builder Notation:

${x | x > - \frac{1}{2}}$

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: $(- \frac{1}{2}, \infty), {x | x > - \frac{1}{2}}$

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

Step 6