Algebra Examples

$f (x) = {\begin{cases} | x | & x \leq 0 \\ \frac{1}{x} & x > 0 \end{cases}$

Step 1

Find the domain.

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Step 1.1

Since the domain of $| x |$ is all real numbers, the domain of this piece of the function is its restriction, $x \leq 0$ .

$(- \infty, 0]$

Step 1.2

Find the domain for $\frac{1}{x}$ .

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

Step 1.3

Compare the domain of $\frac{1}{x}$ with the restriction $x > 0$ to see where this piece of the function is defined.

$(0, \infty)$

Step 1.4

Find the domain by taking the union of all of the intervals where the function is defined.

$(- \infty, \infty)$

Step 2

List the domain and range of the piecewise.

Domain: $(- \infty, \infty)$

Range: $(- \infty, \infty)$

Step 3