If the tangent line to y = f(x) at (4, 3) passes through the point (0, 2), find f(4) and f'(4).

Answer 1

The slope of the tangent line is, from the coordinates of two points you know on the line,
m = (2-3)/(0-4) = 1/4

For the (straight) tangent line,
y = mx + b = (x/4) + b
3 = (4/4) + b
b = 2
Therefore, the tangent line equation is

y = (x/4) + 2

They have already told you that, for the function f(x), f(4) = 3
f'(4) is the slope of the tangent line at x = 4. We have shown that to be 1/4.

You cannot say what the actual function f(x) is. There are an infinite number of possibilities.