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A piecewise function uses different formulas on different intervals of its domain. The absolute value function f(x) = |x| is the simplest example: it equals x when x >= 0 and -x when x < 0. Piecewise functions model real situations where rules change at thresholds (tax brackets, shipping rates, speed limits). They also motivate the concepts of continuity and one-sided limits.
Practice evaluating piecewise functions at specific points, especially at the boundary values. Graph by drawing each piece on its interval, paying attention to open vs. closed endpoints. Discuss continuity informally: does the graph have a break at the boundary?
From your study of function notation and domain/range, you know that a function assigns exactly one output to each input. A piecewise function does this using different formulas on different parts of its domain. The simplest example is the absolute value function: f(x) = x when x ≥ 0 and f(x) = −x when x < 0. There is no contradiction — at any particular input, exactly one formula applies. The function is perfectly well-defined; it just uses different rules in different regions.
Piecewise functions model real situations where rules change at thresholds. Tax brackets are piecewise: your tax rate on the first $10,000 of income differs from the rate on income above $10,000. Shipping costs often jump at weight thresholds. Speed limits change at city boundaries. In each case, the underlying relationship is a single function of one variable, but the formula governing it switches at specific boundary values. Recognizing these as piecewise functions connects abstract function notation to the stepped, threshold-based rules you encounter constantly in everyday life.
Graphing a piecewise function requires attention to three things: drawing each piece on its correct interval, marking boundary points carefully with open or closed circles, and checking whether the pieces connect. An open circle at a boundary means the function does not include that point (strict inequality); a closed circle means it does (inclusive inequality). For f(x) = { x² if x < 2; 3x − 1 if x ≥ 2 }, you draw the parabola y = x² only for x-values strictly less than 2 (open circle at (2, 4)), then the line y = 3x − 1 for x ≥ 2 (closed circle at (2, 5)). The gap between the open and closed circles reveals a jump discontinuity at x = 2.
Not all piecewise functions are discontinuous at their boundaries. If the pieces happen to agree at the boundary — if the left-hand limit, the right-hand limit, and the function value all match — the function is continuous there and the graph passes through the boundary without a break. For example, f(x) = { x if x < 1; 1 if x ≥ 1 } is continuous at x = 1 because both pieces approach 1. This observation previews the formal concept of continuity and one-sided limits that you will study rigorously in calculus. The intuition you build here — checking whether pieces "connect" at boundaries — is exactly what the ε-δ definition of continuity will formalize.