Asked in Math and ArithmeticAlgebraCalculus
How does one determine the validity and the invalidity of a function?
May 13, 2012 3:30PM
The validity or invalidity of a function are not abstract but depend on its domain and codomain or range.
If for any point, A, in the domain there is a unique point, B, in the range such that f(A) = B then the function is valid at A.
The validity of a function can change from point to point.
For example, f(x) = sqrt(x) is not a function from the set of Real Numbers to the set of Real Numbers because any negative number in the domain is not mapped to any value in the range. This can be corrected either by changing the domain to the set of non-negative Real Numbers or (if you are a more advanced mathematician) change the range to the set of Complex Numbers.
Similarly the reciprocal function, f(x) = 1/x is valid everywhere except for x = 0.
Or f(x) = tan(x) is valid except for x = 90+k*180 degrees for all integer values of k - so it is not valid at an infinite number of points.