There are 3 recognisable ways a child goes wrong here — distinct patterns, not carelessness.
Each one below is a mistake our question bank was built to expose, written out the way you would actually hear it said out loud at home.
Knowing which one you are looking at is the whole job. "Check your answer" does not help a child who is confidently applying the wrong rule.
What you might hearI just do the first operation backward first.
The student reverses operations in the order they see them. For f(x) = 3x + 5, they divide by 3 first and then subtract 5, so they get x/3 - 5 instead of (x - 5)/3.
They see the formula as steps to copy from left to right. They do not see it as a chain that must be undone from the outside back.
This one is not a slip to correct — the underlying idea needs re-teaching before more practice will help.
What you might hearI switched x and y, so this is the inverse.
The student swaps x and y and calls the swapped equation the inverse. For y = 2x - 7, they write x = 2y - 7 and stop instead of solving for y.
They think the swap creates the inverse by itself. They miss that the inverse rule must tell the new output.
This one is not a slip to correct — the underlying idea needs re-teaching before more practice will help.
What you might hearInverse means flip it, so I put 1 over it.
The student changes the function into a reciprocal instead of undoing it. For f(x) = x + 4, they say the inverse is 1/(x + 4) rather than x - 4.
They know inverse can mean reciprocal in multiplication. They use that meaning for functions, where inverse means undoing inputs and outputs.
This one responds to correction in the moment — the idea is there, the rule being applied is not.
For each mistake above, Algo School gives you the words to say next — the specific response that corrects the thinking instead of just marking the answer wrong, plus practice aimed at that exact error.
See your child's own results — start freeInverse functions undo each other. The graph shows f(x) = 5x − 10 (solid) and the line y = x (dashed). The inverse f⁻¹ is the reflection of f over y = x. Write an expression for f⁻¹(x).
Answer (x + 10)/5
Generated by build-functions-alg2, a deterministic question engine — not written by an AI, and verified before any child sees it.
Look at what they SAY, not just what they wrote. Each mistake above includes the phrasing a parent typically hears — that sentence is usually the giveaway. In Algo School, the tutor recognises these patterns while your child is working and adapts the session around the one it sees.
Yes. We track 3 distinct approved mistake patterns for this concept, each one written up because it recurs — not because it is unusual. A child making one of them is doing something systematic, which is far easier to fix than random errors.
Respond to the thinking, not the answer. Algo School gives parents the specific wording for each mistake, then generates practice targeted at that error and shows you whether it actually cleared.
They are part of the same content library that drives practice: each one is reviewed and approved before it is used, and the Advanced Function Analysis questions on this site are checked by an automated verification pass before a child is ever served them.
See the full curriculum map — every Common Core concept we cover, the verified question count, and what a child at this level should know.
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