To construct new functions from old ones, we must first rummage through our toolkit to see which functions we already have. What exactly does it mean, though, to "have" a function, to be "given" it, or, for that matter, to "know" it?

Remember that functions are just the rules that tell us how to find unique outputs corresponding to particular inputs. We "give" these rules in various ways:

• Natural Language

  "Take the number of customers, triple it, and add 60. That's how long it will take me to cook for these Visigoths!"

  — temperamental chef "Louis"

  
  

• Equations

  
  
  

• Graphs

  
  
  

• Tables of Values (Data)

  Length of Magnesium Ribbon (x, in centimeters)	Time to Burn (y, in seconds)
  10	13
  20	27
  30	39

  
  
  

These are the starting points for constructing mathematical models. We "have" a function when we have the rule that describes it — in any form that is appropriate or convenient.

In this lesson we'll learn how to take a "given" function and shift it, stretch it, rearrange it, and put it together with other functions so that the end result is something that fits the situation we wish to describe, "just so".

There's an old saying: "If life gives you lemons, make lemonade!" In mathematics, it might go like this: "If you don't have the functions you need, build them!"