How to get the continuous function?

Question

Note - this is similar to Process For Building a Function? but without the emphasis on logarithmic functions in particular.

I have a few different types of graphs that I'd like to express as functions... some of them are curvy, some of them have straight lines. They are all continuous. Here's one example:

For starters, how would I express that shape as a function (e.g. f(t) = ...)?

More fundamentally though, is there a collection of ideas that can be applied to certain classes of problems - like if I see certain shapes it's a quadratic, others it's a bezier function, etc.

I get that this is open-ended, so perhaps there isn't really one answer to accept here - but guidance on how to start thinking more like a mathematician and write solvable functions is much appreciated.

Thanks!

Answer 1

This is a piecewise function, i.e. its definition depends on intervals. In the given case,

$$f(x)=\begin{cases}0\le x\le3\to x,\\x\ge3\to3\end{cases}.$$

If you look closely, you will notice that I allowed equality to $3$ twice, because $f(3)=3$ in both pieces. This stresses that the function is continuous, i.e. the pieces are continuous functions and their values at the common endpoints coincide.

---

Identifying functions is an "art". You need to get acquainted to functions, their shapes and properties. Lookup the keywords "catalog of functions".

Answer 2

You can begin by writing functions piecewise on the interval they are defined. For example, Write above $f(t)$ as

$f(t)=t ;0\leq t \leq 3$

$f(t)=3; t>3$