Question

give a rule of the piecewise - defined function. give the domain and the range

what is the rule? select the correct choice below and fill in the answer boxes within your choice.
a. $f(x)=\

$$\begin{cases} 3 & \\text{if } x < - 1 \\\\ - 2 & \\text{if } x > 2 \\end{cases}$$

$
b. $f(x)=\

$$\begin{cases} \\square & \\text{if } x < \\square \\\\ \\square & \\text{if } x\\geq \\square \\end{cases}$$

$

what is the domain? select the correct choice below and fill in the answer box within your choice.
a. the domain is \\{\\} (use a comma to separate answers as needed.)
b. the domain is $(-\infty, - 1)\cup(2, \infty)$ (type your answer in interval notation.)

what is the range? select the correct choice below and fill in the answer box within your choice.
a. the range is \\{\\} (use a comma to separate answers as needed.)
b. the range is \square (type your answer in interval notation.)

Explanation:

Rule of the Piecewise Function

Step1: Analyze the graph for \( x < -1 \)

From the graph, when \( x < -1 \), the function has a constant value. Observing the left - hand part of the graph (for \( x < - 1\)), the \( y\) - value is 3. So for \( x < - 1\), \( f(x)=3\).

Step2: Analyze the graph for \( x > 2 \)

For the right - hand part of the graph (for \( x > 2\)), the \( y\) - value is - 2. So for \( x > 2\), \( f(x)=-2\).
So the correct rule is \( f(x)=

$$\begin{cases}3& \text{if }x < - 1\\-2& \text{if }x > 2\end{cases}$$

\), so the correct choice for the rule is option A.

Domain of the Function

Step1: Identify the intervals of \( x \)

The function is defined for \( x < - 1\) and \( x > 2\). In interval notation, the domain is the union of the intervals \( (-\infty,-1)\) and \( (2,\infty)\). So the domain is \( (-\infty,-1)\cup(2,\infty)\), so the correct choice for the domain is option B.

Range of the Function

Step1: Identify the \( y\) - values

The function takes only two values: 3 (when \( x < - 1\)) and - 2 (when \( x > 2\)). So the range is the set containing these two values.

Answer:

• Rule: Option A: \( f(x)=

$$\begin{cases}3& \text{if }x < - 1\\-2& \text{if }x > 2\end{cases}$$

\)

• Domain: \( (-\infty,-1)\cup(2,\infty)\) (Option B)
• Range: \(\{ - 2,3\}\) (For the range, since the function only outputs - 2 and 3, the range is the set with these two elements. So for option A of the range, we fill in - 2, 3)