Define and interpret functions with different rules on different intervals.
A piecewise function uses different formulas over different parts of its domain. The absolute value function is a classic example. Piecewise functions model real-world scenarios where rules change based on thresholds (tax brackets, shipping rates, etc.).
Evaluating a Piecewise Function
Using the function above: f(−2) = (−2)² = 4 (first piece). f(1) = 2(1)+1 = 3 (second piece). f(3) = 7 (third piece, since 3 ≥ 3).
Remember This!
Check continuity at the boundary points: compute both one-sided limits and check if they equal the function value. If they differ, there is a jump discontinuity.
For f(x) = {x + 3 if x ≤ 1, x² − 1 if x > 1}, what is f(1) + f(2)?