Find the Domain and Range y = natural log of (-x+3)/2

$y = \ln (\frac{- x + 3}{2})$ ?

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

$\frac{- x + 3}{2} > 0$

Solve for $x$ .

Multiply both sides by $2$ .

$\frac{- x + 3}{2} \cdot 2 > 0 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{- x + 3}{2} \cdot 2 > 0 \cdot 2$

Rewrite the expression.

$- x + 3 > 0 \cdot 2$

Simplify the right side.

Multiply $0$ by $2$ .

$- x + 3 > 0$

Solve for $x$ .

Subtract $3$ from both sides of the inequality.

$- x > - 3$

Divide each term in $- x > - 3$ by $- 1$ and simplify.

Divide each term in $- x > - 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{- 3}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{- 3}{- 1}$

Divide $x$ by $1$ .

$x < \frac{- 3}{- 1}$

Simplify the right side.

Divide $- 3$ by $- 1$ .

$x < 3$

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

Interval Notation:

$(- \infty, 3)$

Set-Builder Notation:

${x | x < 3}$

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

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${y | y \in ℝ}$

Determine the domain and range.

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

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

Find the Domain and Range y = natural log of (-x+3)/2

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