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.
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