Question

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

part 1 of 3

what is the rule? select the correct choice below and fill in the answer boxes within your choice.

a. \\( f(x) = \

$$\begin{cases} \\box & \\text{if } x \\le \\box \\\\ \\box & \\text{if } x > \\box \\end{cases}$$

\\)
b. \\( f(x) = \

$$\begin{cases} \\box & \\text{if } x < \\box \\\\ \\box & \\text{if } x \\ge \\box \\end{cases}$$

\\)

Explanation:

⚡ Using what you learned: Introduction to Piecewise Functions · domain and range

Step 1: Analyze the left piece of the graph

The left piece is a horizontal ray pointing to the left:

• It lies along the horizontal line \( y = -2 \).
• It ends on the right at the point \( (1, -2) \) with a solid (closed) circle, which means \( x = 1 \) is included in this interval.
• Therefore, this piece is defined by:

$$ f(x) = -2 \quad \text{if } x \le 1 $$

Step 2: Analyze the right piece of the graph

The right piece is a horizontal ray pointing to the right:

• It lies along the horizontal line \( y = 3 \).
• It starts on the left at the point \( (1, 3) \) with an open circle, which means \( x = 1 \) is not included in this interval.
• Therefore, this piece is defined by:

$$ f(x) = 3 \quad \text{if } x > 1 $$

Step 3: Match with the given options

Looking at the choices:

• Option A uses the inequalities \( \le \) and \( > \):

$$ f(x) =
LATEXBLOCK0
$$

• Option B uses the inequalities \( < \) and \( \ge \).

Since our intervals are \( x \le 1 \) and \( x > 1 \), Option A is the correct structure.

Answer:

A. \( f(x) =

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

\)