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Imagine ordering custom t-shirts: the first 10 shirts cost 15each,thenext40cost15 each, the next 40 cost 10 each, and any beyond that cost just $6. A single equation cannot describe this pricing model because the rule changes as the quantity grows.
To model situations with changing rules, mathematicians use a piecewise function — a function defined by multiple sub-functions, where each sub-function applies to a specific interval of the domain.
How do you find the output of a piecewise function without getting tangled up in all the different formulas?

Evaluating Branch by Branch

Evaluating a piecewise function is a two-step process: first, test your input xx against each domain condition to find the active branch; second, substitute xx into only that branch's formula.
Consider f(x)f(x) defined as: f(x)={−2x−3if x<−12if −1≤x<2(x−3)2−1if x≥2f(x) = \begin{cases} -2x - 3 & \text{if } x < -1 \\ 2 & \text{if } -1 \le x < 2 \\ (x - 3)^2 - 1 & \text{if } x \ge 2 \end{cases}
To find f(−3)f(-3), check the conditions: since −3<−1-3 < -1, use the first linear rule: f(−3)=−2(−3)−3=6−3=3f(-3) = -2(-3) - 3 = 6 - 3 = 3.
To find f(−1)f(-1), look closely at the boundary: −1-1 satisfies −1≤x<2-1 \le x < 2 (not x<−1x < -1), so the constant branch applies: f(−1)=2f(-1) = 2.
To find f(4)f(4), check the conditions: since 4≥24 \ge 2, use the quadratic rule: f(4)=(4−3)2−1=12−1=0f(4) = (4 - 3)^2 - 1 = 1^2 - 1 = 0.
Evaluating points is straightforward, but what happens when you need to draw the entire picture on a coordinate plane?

Graphing and Boundary Points