First, the dot notation may be confusing. I always have trouble remembering whether fog = f(g(x)) or g(f(x)). If it was written in the same format, maybe the kid would have gotten it. But also, maybe something more fundamental went wrong in the inductive reasoning.

Michael Falk

This kid seems to be adding polynomials rather than composing them, but is also doing that thing kids do where we teach them that two sets of parentheses next to each other means “multiply”, and then we teach them this and are surprised when they multiply.In a one-on-one conference with this student I would rewrite x+7 as 1x+7 and have an explicit discussion of combining like terms vs. multiplying variables, as well as what (f o g)(x) actually means.

maxmathforum

If the challenge were just knowing whether f*g(x) = f(g(x)) or g(f(x)) wouldn’t we have either gotten 4x – 1 + 7 or 4(x + 7) – 1 as the answer?And if the student were just adding polynomials, we would have seen 5x + 6.If they were just multiplying we’d get 4x^2 + 27x – 7, and not two answers… There’s something more special going on in the last line. Did they try to compose both ways to get the two responses 4x^2 – 1 – 7 and 4x^3 + 6? Why get – 1 – 7 one way and +6 another way? Why get 4x^2 once and 4x^3 once?They may have tried doing (4x – 1)(x + 7)(x) and failed to do that at all well, but it seems like they had some idea of applying the operations in one function to the other, but applying the 4x to x and getting 4x^2 and applying the – to the 1 and 7 and getting -1 – 7. It’s not a terrible theory, but it doesn’t get them the answers they got above.How could writing out their work on the composition problems above help them get the symbolic version correct?

mr bombastic

Possibly mis-wrote g(x) =4x – 1 as 4x^2 – 1. Then used f(x) = x + 7 to get to 4x^3 + 6 by multiplying the 4x^2 by x and adding the 7 to the -1. The kid is missing the key point that you are substituting a complicated exression for x. I usually ask things like what is f(DOG).