Precalculus Examples

$f (x) = \ln (3 - x)$

Step 1

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

$3 - x > 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $3$ from both sides of the inequality.

$- x > - 3$

Step 2.2

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

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Step 2.2.1

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.

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

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.2.2.2

Divide $x$ by $1$ .

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

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 3$ by $- 1$ .

$x < 3$

Step 3

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

Interval Notation:

$(- \infty, 3)$

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

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

Step 6