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.

For any nonzero integer k,

y=sinc(k*pi*x/2)

 =(sin(k*pi*x/2))/(k*pi*x/2) for nonzero x,
 
 =1 for x=0

is 1 at x=0 and is zero at when x is any nonzero multiple of 2/k (including 2 and 4). Also, any of these can be multiplied by any  function of the real variable x which equals 1 at x=0, for example by exp(a*x).

See MathWorld or Wikipedia entries for Sinc Function.

Edited on June 19, 2006, 10:58 pm

Posted by Richard on 2006-06-19 22:35:30