Functions at the 700-800 Level

Hard-tier function items are chains, not evaluations. A stem gives two or three rules in different units — pages per minute, dollars per page, kilograms per pound, gallons per mile — and the difficulty lives entirely in which link of the chain the last sentence names.

Four archetypes cover almost everything. The rate chain: composition disguised as a word problem, usually run backward from a cost or a budget, with a fixed fee planted so that dividing the whole output by the rate produces a clean wrong answer. Table reversal: $f(g(x)) = k$ solved outside-in, where the value of $g(x)$ is the most-chosen distractor. Transformations: inside changes the input and does the opposite of what it looks like; outside changes the output; and a reflection applied after a shift negates everything already there. Inverses: undoing a composition reverses the order, so a budget becomes $h^{-1}(c^{-1}(B))$, never $c^{-1}(h^{-1}(B))$.

The distractors are systematic rather than random: the intermediate value in the wrong units, the dropped fixed fee, the division by a rate from the wrong link, and the direction error. This module trains you to write the chain with units before computing, to collapse multi-rate chains into one constant, and to spend the last twenty seconds auditing the units of your answer against the units of the ask.