Notation for a set of functions with different parameter values

Question

I have some functions $f_1(x), f_2(x), ... f_n(x)$ which all are of the same type. Lets say all are linear functions with $f(x)=mx+b$ but with different values for $m$ and $b$.

More generic, I have parameters $p_1, p_2, ... p_k$ and more complex functions $f(x;p_1, p_2, ...p_k)$ that I don't want to write out (since they are the result of some iterative scripts and quite complex).

What I want to find is a notation for these generic functions. So a notation to describe that a set of functions shares the same function type but each function has different parameters without explicitly writing the function out.

Best, Julz

Answer 1

Using your linear function example, how about $f(x;m,b)=mx+b$. Then you can write $\theta=(m,b)$ as the parameter and write out your functions as $f(x;\theta_1)=m_1x+b_1, f(x;\theta_2)=m_2x+b_2,\cdots$

Answer 2

Writing the parameters as a vector $p=(p_1, ..., p_k)$ we also often write $f_p$. If you want to consider a sequence of functions $f_1, f_2,...$, with parameter vectors $p^{(1)}, p^{(2)}, ...,$ you might then also write them as $f_{p^{(1)}}, f_{p^{(2)}},...$