Give a rule for each piecewise-defined function. Also give the domain and range.

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Step 1: Identify the two pieces of the piecewise function from the graph. For x ≥ 0, the function is a horizontal line at y = 5, but the point at (0, 5) is an open circle, meaning it is not included in the function at x = 0. For x < 0, the function is a horizontal line at y = -3, and the point at (0, -3) is a closed circle, meaning it is included in the function at x = 0.

Step 2: Write the rule for the piecewise function based on the observations: For x ≥ 0, y = 5 (excluding x = 0), and for x < 0, y = -3 (including x = 0). This can be expressed as: \(f(x) = \begin{cases} -3 & \text{if } x \leq 0 \\ 5 & \text{if } x > 0 \end{cases}\)

Step 3: Determine the domain of the function. Since the function is defined for all x-values on the number line (both left and right of zero), the domain is all real numbers, which can be written as \((-\infty, \infty)\).

Step 4: Determine the range of the function. The function only takes on the values -3 and 5, so the range is the set containing these two values: \({-3, 5}\).

Step 5: Summarize the piecewise function rule, domain, and range clearly: The function is \(f(x) = \begin{cases} -3 & x \leq 0 \\ 5 & x > 0 \end{cases}\), with domain \((-\infty, \infty)\) and range \({-3, 5}\).

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주요 개념

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Piecewise-Defined Functions

A piecewise-defined function is a function composed of multiple sub-functions, each applying to a specific interval of the domain. Understanding how to write the rule for each piece based on the graph is essential for describing the function accurately.

Domain and Range

The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). Identifying these from a graph involves noting the intervals over which the function is defined and the corresponding output values.

Interpreting Graphs with Open and Closed Points

Open points on a graph indicate that the function does not include that point, while closed points indicate inclusion. This distinction affects the domain and range and is crucial when writing piecewise functions to specify where each rule applies.