Question

2.6.35
3 parts of 1 point

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) = \

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

\\)

b. \\( f(x) = \

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

\\)

Explanation:

🆕 New Concept Discovered: Piecewise-Defined Functions
Different rules for different x-values

Step 1: Analyze the graph for the first interval

Looking at the graph, the grid lines represent intervals of \(1\) unit.

There is a horizontal line on the left side:

• It lies at \(y = -3\).
• It extends to the left indefinitely, meaning \(x\) goes to \(-\infty\).
• On the right end of this segment, at \(x = 1\), there is a solid (closed) circle at the point \((1, -3)\).
• A solid circle means the value \(x = 1\) is included in this interval, which corresponds to the inequality \(x \le 1\).

So, the rule for this part is:

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

Step 2: Analyze the graph for the second interval

There is another horizontal line on the right side:

• It lies at \(y = 2\).
• On the left end of this segment, at \(x = 1\), there is an open circle at the point \((1, 2)\).
• An open circle means the value \(x = 1\) is not included in this interval, which corresponds to the inequality \(x > 1\).
• The line extends to the right indefinitely, meaning \(x\) goes to \(\infty\).

So, the rule for this part is:

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

Step 3: Match with the given options

Comparing our findings with the choices:

• Option A uses the inequalities \(x \le 1\) and \(x > 1\), which matches our boundary point at \(x = 1\) and the closed/open circle positions.
• We fill in the function values:
• \(f(x) = -3\) if \(x \le 1\)
• \(f(x) = 2\) if \(x > 1\)

Answer:

Select A and fill in the boxes as follows:

$$ f(x) =
LATEXBLOCK0
$$