o Choosing a Branch
-> A piecewise function uses different rules on input intervals.
-> Choose the branch by the input, not by the output.
-> First compare x with each branch's interval condition.
-> Substitute into only the rule whose condition holds.
-> An included input must have exactly one output.
-> Check boundary inputs carefully for < versus <=.
-> An input outside all listed intervals is outside the domain.
-> A branch condition may test x, or it may name an interval.
-> With x >= 2 listed, every larger x is already covered.
o Worked Piecewise Example
-> Illustrative example using three distinct input intervals.
-> Define f(x) = 2x for x < 0.
-> Define f(x) = 3 for 0 <= x < 2.
-> Define f(x) = -x + 5 for x >= 2.
-> f(-2) = 2(-2) = -4 using the first branch.
-> f(1) = 3 using the middle branch.
-> f(4) = -4 + 5 = 1 using the last branch.
-> f(0) uses the middle branch because 0 satisfies 0 <= 0.
-> f(2) uses the last branch because 2 satisfies x >= 2.
o Restricted Graphs
-> Graph each rule only on its assigned input interval.
-> For the example, draw y = 2x only to the left of x = 0.
-> Mark (0, 0) open; mark (0, 3) closed.
-> Draw y = 3 from x = 0 up to, but not including, x = 2.
-> The middle branch has an open endpoint at (2, 3).
-> The last branch includes (2, 3); the final point is closed.
-> Thus f(0) = 3 and f(2) = 3, each with one output.
-> f is increasing, flat, then decreasing across its domain.
o Jersey Total Cost
-> Let n be the number of jerseys; C(n) is total cost.
-> The domain is positive integers, not all real numbers.
-> C(n) = 88n for 1 <= n <= 5.
-> C(n) = 75n for 6 <= n <= 11.
-> C(n) = 65n for 12 <= n <= 29.
-> C(n) = 56n for n >= 30.
-> These are all-units discounts: one rate for the whole order.
-> The tiers leave no gap: 1-5, 6-11, 12-29, then 30 and up.
-> Reading the domain first stops you from using 0 jerseys.
o Jersey Calculations
-> For 11 jerseys, choose the 6 through 11 tier.
-> C(11) = 75(11) = 825 dollars.
-> For 25 jerseys, choose the 12 through 29 tier.
-> C(25) = 65(25) = 1625 dollars, not 1885.
-> C(5) = 440, while C(6) = 450.
-> C(11) = 825, while C(12) = 780: total cost can fall.
-> C(29) = 1885, while C(30) = 1680.
-> Rates step down: 88, 75, 65, then 56 per jersey.
-> At n = 30 the total drops the most: 205 dollars.
o Rates and Model Checks
-> An all-units price changes for every item at a tier boundary.
-> A progressive rate applies each rate only to its own band.
-> Do not add earlier tier costs in this all-units model.
-> Unit price is constant within tiers and jumps between them.
-> Total cost rises within tiers; it is not a step function.
-> Plot separate integer points; do not fill gaps in the domain.
-> Tier prices jump; the integer-domain total has tier changes.
-> Compare the two models at n = 12 before choosing one.