f (x) and g(x) are two quadratic functions such that f (1) – g(1) = 1, f (2) – g(2) = 2 and f (3) – g(3) = 5. Find the value of f (4) – g(4)?

To solve this problem, we need to understand that the difference between two quadratic functions is also a quadratic function. Let's denote h(x) = f(x) - g(x).

Given that h(1) = 1, h(2) = 2, and h(3) = 5, we can see that h(x) is a quadratic function that passes through the points (1,1), (2,2), and (3,5).

A quadratic function is of the form ax^2 + bx + c. We can set up a system of equations to find the coefficients a, b, and c using the points given:

For x=1, y=1: a + b + c = 1 For x=2, y=2: 4a + 2b + c = 2 For x=3, y=5: 9a + 3b + c = 5

Solving this system of equations, we find that a = 1, b = -1, and c = 1. So, h(x) = x^2 - x + 1.

Finally, to find the value of f(4) - g(4), we just need to find h(4) = 4^2 - 4 + 1 = 13.