If is a set and are functions then, are always functions, in general distinct from and . Eventually could appear or again. For example if for all then, .
My textbook just gave the definitions (f+g)(x)=f(x)+g(x)
and also for subtraction and division. Will these operations always produce new functions or is this notation only applicable when it does? If it always will, how can you prove it?
Every function operation has an "identity". For sum and difference, that identity is the function that is identically 0.
If f(x) is any function and g(x)= 0 for all x, then (f+ g)(x)= f(x)+ g(x)= f(x)+ 0= f(x) and (f- g)(x)= f(x)- g(x)= f(x)- 0= f(x).
For multiplication and division, that identityis the function that is identically 1.
If f(x) is any function and g(x)= 1 for all x, then (fg)(x)= f(x)g(x)= f(x)(1)= f(x) and (f/g)(x)= f(x)/g(x)= f(x)/1= f(x).
For composition, the identity is the function g(x)= x, which is called the "identity function".
If f(x) is any function and g(x)= x, and .
The fact that the result of these operations are functions themselves now makes sense to me for the following logic:
If for each input (x) you get your new output[f+g(x)] by adding the outputs [f(x), g(x)] then because the result of addition is a unique number there cannot possibly be two different outputs. (Thus it is still a function by our definition of a function.)
The logic is exactly the same for subtraction, multiplication and division (provided g(x) is not 0).
When one composes f(g(x)) then as long as the range of g(x) was in the domain of f(x) then one can be guaranteed a unique value for any x.
However, is this proved in any college classes perhaps?