The discussion revolves around whether two different functions can yield the same output for a specific input point. Participants explore this question in the context of function definitions, domains, and the concept of bijections.
Participants express differing views on the implications of functions having the same output at a point while being distinct. There is no consensus on the broader implications of this scenario, as some examples suggest it is common, while others emphasize the importance of domain differences.
Participants note that the discussion hinges on the definitions of functions and their domains, which may affect the interpretation of when two functions can be considered the same or different.
This happens all the time: ##p=0## for ##f(x)=x^2\; , \;g(x)=\sin(x)\; , \;h(x)=|x|## etc. However ##f=g## if ##f(p)=g(p)## for all points ##p \in \operatorname{dom}(f)=\operatorname{dom}(g)##. We also widely use the fact, that functions are the same at one point and close in their neighborhood, when we approximate a function by its tangent in ##p##.kent davidge said:Today, while reading about bijections, a question came into my mind. And that is: is there any way that two different functions ##f## and ##g## acting on a same point ##p## gives the same output? In symbols, as I'm not good in English, is it possible that ##f (p) = g(p)## with ##f \neq g##?
Here's an example that's almost what you're talking about:kent davidge said:Today, while reading about bijections, a question came into my mind. And that is: is there any way that two different functions ##f## and ##g## acting on a same point ##p## gives the same output? In symbols, as I'm not good in English, is it possible that ##f (p) = g(p)## with ##f \neq g##?