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

Verified step by step guidance

1. Step 1: Identify the two pieces of the piecewise function based on the graph. The graph shows two horizontal lines: one starting at (0, 5) and extending to the right, and another starting at (0, -3) and extending to the left.
2. Step 2: Write the rule for the right piece (x ≥ 0). Since the line is horizontal at y = 5, the function rule for this piece is \(f(x) = 5\) for \(x \geq 0\). Note that the point at (0, 5) is an open circle, so the value at x = 0 is not included in this piece.
3. Step 3: Write the rule for the left piece (x < 0). The line is horizontal at y = -3, and the point at (0, -3) is a closed circle, so the function rule for this piece is \(f(x) = -3\) for \(x < 0\).
4. Step 4: Define the domain of the piecewise function. Since the function covers all x-values from negative infinity to positive infinity, the domain is \((-\infty, \infty)\).
5. Step 5: Define the range of the function. The function only takes the values 5 and -3, so the range is the set \({-3, 5}\).

Key Concepts

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 involves identifying the function's behavior on different parts of the x-axis, often using inequalities to specify the domain for each piece.

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). For piecewise functions, the domain is often split into intervals, and the range is determined by the output values of each piece over its domain.

Interpreting Graphs with Open and Closed Points

Open and closed points on a graph indicate whether a function includes or excludes a point at a boundary. A closed point means the function value is included at that x-value, while an open point means it is excluded. This distinction is crucial for correctly defining the domain intervals and function values in piecewise functions.