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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:

enter image description here

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!

davidkomer
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2 Answers2

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

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

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

User
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Pi_die_die
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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".

  • Thanks! Now I'm wondering... if we make the beginning of it start off with a curve - that would simply change the first piece (to be some operations with x)? – davidkomer Jul 26 '18 at 06:51
  • @davidkomer: try something. –  Jul 26 '18 at 07:23