2011-05-18, 05:09 PM
This question is conceptual. All throughout my young'un days, I was told that "functions must be one-to-one". All curves that do not pass the "vertical line test" are not functions. All curves that do not pass the "horizontal line test" do not have inverses.
Why do functions have to be "one to one"? I don't understand the purpose of "not having two outputs for a single input". If I want to visualize the plot of f(x)=sin(x), I would plot all possible points in the domain t=[-∞,∞] for (x,y)=(t,sin(t)). I can also visualize reflections over the line y=x (and equivalently x=y), by plotting all possible points in the domain t=[-∞,∞] by d(t)=(sin(t),t). Yet supposedly, d(t)=(sin(t),t) is not a function (and consequently not an "inverse") because it "doubles-back on itself".
x=sin(y) cannot possibly be a function; no... no way.
Why do functions have to be "one to one"? I don't understand the purpose of "not having two outputs for a single input". If I want to visualize the plot of f(x)=sin(x), I would plot all possible points in the domain t=[-∞,∞] for (x,y)=(t,sin(t)). I can also visualize reflections over the line y=x (and equivalently x=y), by plotting all possible points in the domain t=[-∞,∞] by d(t)=(sin(t),t). Yet supposedly, d(t)=(sin(t),t) is not a function (and consequently not an "inverse") because it "doubles-back on itself".
x=sin(y) cannot possibly be a function; no... no way.