Find the Domain and Range y = natural log of |x|

$y = \ln (| x |)$

Step 1

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

$| x | > 0$

Step 2

Solve for $x$ .

Write $| x | > 0$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

In the piece where $x$ is non-negative, remove the absolute value.

$x > 0$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x > 0$

Write as a piecewise.

${\begin{cases} x > 0 & x \geq 0 \\ - x > 0 & x < 0 \end{cases}$

Find the intersection of $x > 0$ and $x \geq 0$ .

$x > 0$

Divide each term in $- x > 0$ by $- 1$ and simplify.

Divide each term in $- x > 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{0}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{0}{- 1}$

Divide $x$ by $1$ .

$x < \frac{0}{- 1}$

Simplify the right side.

Divide $0$ by $- 1$ .

$x < 0$

Find the union of the solutions.

$x < 0$ or $x > 0$

Step 3

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

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: $(- \infty, 0) \cup (0, \infty), {x | x \neq 0}$

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

Step 6

Find the Domain and Range y = natural log of |x|

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