Given the information that the graph of a function has a y-intercept at (0,1) and exactly two x-intercepts at (2,0) and (4,0), how many different functions can you find that pass through these three points?
Note: there are infinite families of functions such as high degree polynomials which pass through them, so a single example would suffice for them. Also disallowed would be piecewise function and functions with artificially restricted domains.
We can always multiply Mindy's function, or Joe's function, by
exp(a*x), or by any other function that is 1 at x=0 and has no zeros,
such as 1/(1+x^2). I am assuming that the restrictions stated mean that
the answer has to be defined at each real x, and that we can't just
define the answer function over the interval [-1,5], say, where
sinc(pi*x/2) has only 2 and 4 as zeros. The point of this problem, if
any, eludes me. Perhaps the official solution will be enlightening.
Posted by Richard
on 2006-06-20 16:13:53