Do any two different functions exist that cannot be simplified to the same form but generate the same graph?
In: Mathematics
Yes if the functions are different in dimensions you are not graphing.
For example, a disc and a sphere might both graph a circle in 2D but differ in a third dimension.
There are piecewise functions that have names, but you can’t really go from the piecewise definition to the named function without knowing what it is. For example, there’s no algebraic way to go from f(x) = {-x for x<0, x for x>=0} to f(x) = absolute value of x, but they’re the same function by definition.
Yes, there are, but I can not give you an example. You can show that there are theorms that are true but impossible to prove. You could take a function that is equal to 1 iff such a theorm is true for some value. This function is equal to 1, but you can not show that.
No. The mathematical definition of a graph of a function f is the set of ordered pairs (x,f(x)) for all x in the domain. If you have another graph (x,g(x)), and you can show that (x,f(x))=(x,g(x)), then this can be simplified to show that f(x)=g(x).
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