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.
\\(\bigcirc\\) a. \\(f(x)=\

$$\begin{cases}\\square & \\text{if } x\\leq\\square \\\\ \\square & \\text{if } x>\\square \\end{cases}$$

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

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

\\)

Explanation:

Step1: Analyze the Graph

From the graph, we can see that there is a horizontal line at \( y = 1 \) for \( x < - 1 \) (assuming the vertex is at \( x=-1 \)) and a horizontal line at \( y = - 1 \) for \( x\geq - 1 \)? Wait, no, looking at the grid, the left part (arrow going left) is at \( y = 1 \) and the right part (arrow going right) is at \( y=-1 \)? Wait, maybe I misread. Wait, the graph: let's assume the point is at \( x = - 1 \), \( y = 1 \) on the left segment and \( y=-1 \) on the right? Wait, no, the standard piecewise function with two horizontal lines. Let's re - examine.

Wait, the graph has two horizontal segments. Let's assume the break point is at \( x=-1 \). For \( x < - 1 \), the function value is \( 1 \), and for \( x\geq - 1 \), the function value is \( - 1 \)? Wait, no, maybe the other way. Wait, the left arrow (as \( x\to-\infty\)) is at \( y = 1 \), and the right arrow (as \( x\to\infty\)) is at \( y=-1 \), with a point at \( x = - 1,y = 1 \) (closed circle) and open circle? Wait, the options are A: \( f(x)=

$$\begin{cases}\square&\text{if }x\leq\square\\\square&\text{if }x>\square\end{cases}$$

\) and B: \( f(x)=

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

\)

Let's assume the break point is at \( x=-1 \). Let's say for \( x < - 1 \), \( f(x)=1 \), and for \( x\geq - 1 \), \( f(x)=-1 \). So the rule would be \( f(x)=

$$\begin{cases}1&\text{if }x < - 1\\-1&\text{if }x\geq - 1\end{cases}$$

\), which matches option B.

Step2: Determine the Domain and Range

• Domain: The domain of a piecewise function is the set of all real numbers for which the function is defined. Since the function is defined for all real numbers (the left segment goes to \( -\infty \) and the right segment goes to \( \infty \)), the domain is \( (-\infty,\infty) \) or \( \mathbb{R} \).
• Range: The range is the set of all output values. The function takes two values, \( y = 1 \) and \( y=-1 \), so the range is \( \{ - 1,1\} \)

For the rule:
For option B, \( f(x)=

$$\begin{cases}1&\text{if }x < - 1\\-1&\text{if }x\geq - 1\end{cases}$$

\)

Answer:

The rule is \( f(x)=

$$\begin{cases}1&\text{if }x < - 1\\-1&\text{if }x\geq - 1\end{cases}$$

\) (corresponding to option B: \( f(x)=

$$\begin{cases}\boldsymbol{1}&\text{if }x<\boldsymbol{-1}\\\boldsymbol{-1}&\text{if }x\geq\boldsymbol{-1}\end{cases}$$

\))

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

Range: \( \{ - 1,1\} \)