Algebra Examples

f (x) = {\begin{cases} x + 1 & x > 1 \\ x - 2 & x \leq 1 \end{cases}

$f (x) = {\begin{cases} x + 1 & x > 1 \\ x - 2 & x \leq 1 \end{cases}$

Step 1

Find the domain.

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Step 1.1

Since the domain of

x + 1

$x + 1$ is all real numbers, the domain of this piece of the function is its restriction,

x > 1

$x > 1$ .

(1, \infty)

$(1, \infty)$

Step 1.2

Since the domain of

x - 2

$x - 2$ is all real numbers, the domain of this piece of the function is its restriction,

x \leq 1

$x \leq 1$ .

(- \infty, 1]

$(- \infty, 1]$

Step 1.3

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

(- \infty, \infty)

$(- \infty, \infty)$

(- \infty, \infty)

$(- \infty, \infty)$

Step 2

The range is the set of all valid

y

$y$ values. Use the graph to find the range.

(- \infty, - 1] \cup (2, \infty)

$(- \infty, - 1] \cup (2, \infty)$

Step 3

List the domain and range of the piecewise.

Domain:

(- \infty, \infty)

$(- \infty, \infty)$

Range:

(- \infty, - 1] \cup (2, \infty)

$(- \infty, - 1] \cup (2, \infty)$

Step 4

f (x) = {\begin{matrix} x + 1 & for & x > 1 \\ x - 2 & for & x \leq 1 \end{matrix}

$f (x) = {\begin{matrix} x + 1 & for & x > 1 \\ x - 2 & for & x \leq 1 \end{matrix}$

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