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.
There is an infinite number of polynomials which satisfy the conditions for the intercepts.
The quadratic polynomial f(x) = (1/8)x^2 - (3/4)x + 1 = 0 is the unique quadratic which passes through the three points given. Let g(x) be any even degree polynomial which is strictly positive and passes through (0,1). Then f(x)*g(x) is also an answer to the problem.
For example, g(x) = (1/32)x^2 + (1/4)x + 1. Then f(x)*g(x) = (1/256)x^4 + (1/128)x^3 - (1/32)x^2 - (1/2)x + 1 is an answer.