How do you do this college level algebra question?

Question

If the function f satisfies the equation f(x + y) = f(x) + f(y) for every pair of real numbers x
and y, what are the possible values of f(0) ?
A. Any real number
B. Any positive real number
C. 0 and 1 only
D. 1 only
E. 0 only


Can anybody show me how to do it? I know the answer, but I don't know how to do it.

Answer 1

Okay, the answer is A because the definition of f(x+y) is f(x)+f(y) therefore no matter what value for X and Y, the equation remains true because they are the same thing.

Example:
For Function F ->
X=5 Y=10

F(x+y)= 5+10= 15
F(x) + F(y)= 5+10= 15
15=15
All Real Numbers

Answer 2

If the domain is all real numbers, I would say (A.) any real number.
You're looking for the value of the function, not the variables. The x and y are variables (or points on the x axis). f(x+y), or f(0), is the function(the line on the y-axis).
If you're only adding (or subtracting) it will always be any real number, be it a fraction, decimal... whatever you want it to be- it will always be any real number.

For instance, f(0)=f(0)+f(0), where f(c)=3c+2. C is either x or y. f(0)=3(0)+2+3(0)+2= 2+2= 4
The function f(0), in this case, has a value of 4

As long as the function isn't multiplying, dividing, of undefined, it's any real number.