Advanced Math/Composition of Functions
QUESTION: Express the function f(x) = 1/(x+1) in the form goh. If h(x) = x+1, find the function g(x).
f(x) = goh
1/(x+1) = g[h(x)]
1/(x+1) = g[x+1]
...And that is where I am stuck, I do not know what to do next to obtain the function g(x). I was thinking perhaps inversing the g[x+1] to g^-1[x+1], but I am unsure of how to continue on the other side of the equation.
ANSWER: Defining h(x) = 1+x is a good start.
Now we want to end up with f(x) = 1/(x+1) which would be f(x) = 1/h(x). This would define the function g by g⚬h = 1/h.
When I work with compositions of functions, I think in terms of the notation g⚬h = g(h), which, as you can see, works pretty well for the above example.
---------- FOLLOW-UP ----------
QUESTION: Thank you! However, could you possibly show the systematic order of operations continuing from where I originally left off? It would be easier for me to make sense of how the solution was obtained.
I think your conceptual difficulties arise from trying to define the function g in terms of x instead of h. Let me try to clarify.
If I have a function, say g, it operates on an argument, which I'll take as t for now so as not to confuse things with the variable x; thus we have g(t). The function g represents a 'rule' for how to manipulate its argument t. For example
g(t) = 1/t
or g(t) = t^2
or any rule (transformation) that represents a function (remember, technically, a function is a transformation that can assign only one value to a given value of its argument, but don't let that sidetrack you too much if it doesn't make sense immediately).
Getting back to the original problem, you want to define g as a function of h, not x (not explicitly at this point). So we have
f(x) = g(h(x)) = 1/h = 1/(1+x)
which in words says " f is a function of x and is the same as g as a function of h which is a function of x".
As another example, let f(x) = (1+x)^2. This could be written as g(h) = h^2 = (1+x)^2 where you just substitute h = (1+x).
Thus you can think of the argument of g as an independent variable without having to explicitly show the argument's dependence on another variable, say x. Another way to say this is that, in these types of problems, the variable x is 'an argument of an argument', which is where the term "composition" comes from.
This concept of treating the argument of a function just like you would as if were a variable is a key concept when you get to calculus and need to take derivatives. A rule known as the Chain Rule allows you to take derivatives of arguments separately.