Inverse Functions
▶︎ Watch it animatedinteractive step-through · ~3 min · optional ⚙︎ Open the appletInverse Lab · the reflection and the reciprocal, side by sideAn inverse function runs the original backward: every input-output pair of f, reversed. Tables invert by swapping columns, graphs by reflecting over the line y = x, and rules by solving for the input. The defining check is composition: do the process, undo it, land where you started. And the notation is a known hazard: the superscript minus one names the inverse, never a reciprocal.
Undoing also swaps roles wholesale: the domain of the inverse is the range of the original, and the range is the original domain. When outputs repeat, no inverse function exists until the domain is restricted to a one-to-one piece. Those two ideas, the reciprocal misread and the swap that students skip, are the misconceptions this topic drills.
The work
3 ways in · any order
Lesson
Inverse Functions
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The inverse undoes the process, pair by reversed pair. The lesson breaks the reciprocal misreading with numeric checks, makes the domain-range swap automatic, and handles the one-to-one requirement with restrictions. Ten scenarios: find, evaluate, and verify inverses from rules, tables, and points.
Diagnostic
10-item topic check
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Ten items spanning the two Topic 2.8 misconceptions: reading f inverse as one over f, and losing the domain-range swap or the one-to-one requirement. Results route you to the drills that fix what fired.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.