The equation f(x+y)=f(x)+f(y)+1 is used to determine the relationship between the values of a function for two different inputs (x and y) and the sum of those inputs.
f(1)=2 is given as an initial condition or "starting point" for the function. It tells us that when the input is 1, the output of the function is 2.
We can use the equation f(x+y)=f(x)+f(y)+1 to manipulate the given information. By plugging in x=1 and y=2, we get f(3)=f(1)+f(2)+1. Since we know that f(1)=2, we can substitute that in and get f(3)=2+f(2)+1. This means that in order to find the value of f(3), we need to determine the value of f(2).
Using the equation f(x+y)=f(x)+f(y)+1, we can plug in x=1 and y=1 to get f(2)=f(1)+f(1)+1. Since we know that f(1)=2, we can substitute that in and get f(2)=2+2+1=5.
Using the value we found for f(2) in the previous question, we can substitute that into our equation for f(3), f(3)=2+f(2)+1=2+5+1=8. Therefore, the final answer for f(3) is 8.