Can understand the solution!

Let the function g(a, b) = f(a) + f(b).

For which function f below will g(a + b, a + b) = g(a, a) + g(b, b)?

A: x+3

B: x^2

C: |x|

D: 1/x

E: x/4

We can plug in values.
Since one of the answer choices is |x|, we should plug in at least one negative value.
Let a=-2 and b=3.
Then a+b = -2+3 = 1.

The question becomes:
For which function f below will g(1, 1) = g(-2, -2) + g(3, 3)?

Since g(a,b) = f(a) + f(b):
g(1,1) = f(1) + f(1) = 2f(1)
g(-2,-2) = f(-2) + f(-2) = 2f(-2)
g(3,3) = f(3) + f(3) = 2f(3)

Substituting these relationships into g(1,1) = g(-2,-2) + g(3,3), we get:
2f(1) = 2f(-2) + 2f(3)
f(1) = f(-2) + f(3).

Thus, the question now becomes:
For which function f below will f(1) = f(-2) + f(3)?

A: x+3
f(1) = 1+3 = 4.
f(-2) + f(3) = (-2+3) + (3+3) = 10.
Doesn't work. Eliminate A.

B: x²
f(1) = 1² = 1.
f(-2) + f(3) = (-2)² + 3² = 13.
Doesn't work. Eliminate B.

C: |x|
f(1) = |1| = 1.
f(-2) + f(3) = |-2| + |3| = 5.
Doesn't work. Eliminate C.

D: 1/x
f(1) = 1/1 = 1.
f(-2) + f(3) = -1/2 + 1/3 = -1/6.
Doesn't work. Eliminate D.

The correct answer is E.

E: x/4
f(1) = 1/4
f(-2) + f(3) = -2/4 + 3/4 = 1/4.
Success!

Just to chime in with a technical approach:

Let's use the definition of the function first.

g(a+b,a+b) = g(a,a) + g(b,b)

is really

f(a+b) + f(a+b) = f(a) + f(a) + f(b) + f(b)

is really

2f(a+b) = 2f(a) + 2f(b)

is really

f(a+b) = f(a) + f(b)

At this point, note that if "f" represents multiplication, f(a+b) always = f(a) + f(b). (Another way of seeing this: x*(a+b) = x*a + x*b.)

So any function that gives us multiplication by a constant should do it. (E) gives us f(x) = (1/4) * x, so f here represents multiplication by a constant, 1/4, and that makes it our answer!