I was attempting to answer this rather-logical problem
A certain function f has the property that *f (x + y) = f (x) + f (y) * for all values of x and y. Which of the following statements must be true when a = b?
i. f (a+b) = 2 f(a)
ii. f (a+b) = [ f (a) ]^2
iii. f(b) + f(b) = f (2a)
f(2a) is the value of f applied at 2a. 2f(a) is two times the value of f applied at a.
For example, if f(x) = x^2, then f(2a) = (2a)^2 = 4a^2, and 2f(a) = 2a^2.
But if you want a more literal answer, the difference between f(2a) and 2f(a) is zero.
In this specific problem you can illustrate the equality by substitution.
let x = a and let y = b
f(a+b) = f(a)+f(b)
a = b so put in a where there is b
f(a+a) = f(a)+f(a)
simplify
f(2a) = 2f(a)
Another interesting aspect of this problem is that there are certain requirements for it to hold true. if we derive both sides with respect to a we get 2f’(2a) = 2f’(a) or f’(2a) = f’(a). This requires that for a function f(ca) to equal cf(a), where c is some constant, the derivatives of them must be the same. This is met by linear functions which all have a derivative of 0 (if you can find another type of function which has a constant derivative which can be modified to meet these characteristics please share.)
^ Your second observation is probably not in th SAT subject test scope.
Oh, that observation is most certainly not within the scope of the test. I just mentioned it because it was interesting and semi-related to the question. It illustrates what conditions act as the line between no difference and difference rather than just explaining this specific situation.
@therockmorten I don’t understand your “out-of-scope” observation. How do you know f is differentiable, or even continuous? Also, a is a constant in this case, so it doesn’t make much sense to take a derivative with respect to a.
lol I only use calculus to find max/min extrema.
@MITer94 I do not know if f is differentiable. I made my comment for differentiable functions which satisfy the characteristics I mentioned. With regards to a being a constant, that is the only type of function i can quickly think of where the statement holds true. I just used a because it was a letter which I was recently thinking about. It can be f of whatever variable as long as it equals a constant. There is a reason I added very specific stipulations to my statement holding true.
If f is differentiable everywhere, then it should be valid to say that f’(2x) = f’(x) for all x. But this is a weaker statement, as all linear equations with a defined slope fit this statement, but only equations of the form f(x) = mx satisfy the original functional equation (f(x) = mx + b with b nonzero isn’t a solution).
However the original functional equation is quite tricky so here is some more info on it:
https://en.m.wikipedia.org/wiki/Cauchy%27s_functional_equation