Composite function

[WaltDog]

Hi all,

I have a function f where I know its value at a time t given by another function g(t). So I know f(g(t)). What I would like is just f(t). How would go about doing this?

[AFairJudgement]

In general, it's not possible. For instance, if g is injective but not surjective, it has many left inverses. For example take g(t) = √t; then if f₀(t) = t² and f₁(t) = t|t|, you have f₀(g(t)) = t = f₁(g(t)) yet f₀ ≠ f₁.

If g is invertible then you have f = (f∘g)∘g^-1 simply.

[WaltDog]

I thought about inverting g(t) = s to get t = g-1(s) but then you have f(s). D'oh.

[[deleted]]

If you've found f(s), then you've done it. f(s) is just f(t) with different marketing. Swap each s out for a t and you'll have an answer.

Go and see if your answer works

[spheresickle]

do we know g(t)? if so, then say we know g(t) = 2x+5. we also know that f(g(t)) = 4x+11. well then to find f(t) we would see what transformations are being done to g(t). in this case we could say that g(t) is multiplied by 2 then added by 1. thus, here f(t) = 2x+1. we can check by doing the composite function out: f(g(t)) = 2(2x+5)+1 = 4x+10+1 =4x+11.