For which of the following does f(a)−f(b)=f(a−b) for all values of a and b?

A) f(x)=x^2
B) f(x)=x/2
C) f(x)=x+5
D) f(x)=2x−1
E) f(x)=|x|
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For which of the following does f(a)−f(b)=f(a−b) for all values of a and b?
f(x)=x^2
f(x)=x/2
f(x)=x+5
f(x)=2x−1
f(x)=|x|

OE

Since, via the distributive property, x(a + b) = ax + bx, you can safely say that any division or multiplication will not change the value of f(a - b) equalling f(a) - f(b). For any of the others, however, there exist many values for which there will be different results.

Another way to look to this question

let a=3 & b=2

Question becomes : For which of the following functions does f(3) - f(2) = f(1)?

Answer choice A:
f(3) = 9
f(2) = 4
f(1) = 1
9 - 4=5........Eliminate

Answer choice B:
f(3) = 3/2
f(2) = 1
f(1) = 1/2
3/2 - 1= 1/2........Keep

Answer choice C:
f(3) = 8
f(2) = 7
f(1) = 6
8 - 7= 1........Eliminate

Answer choice D:
f(3) = 5
f(2) = 3
f(1) = 1
5 - 3= 2........Eliminate

Answer choice E:
f(3) = 3
f(2) = 2
f(1) = 1
3 - 2= 1........Keep

So we can choose another example. Let a=2 & b=3

Answer choice E:
f(2) = 2
f(3) = 3
f(1) = 1
2 - 3= -1.......Eliminate

only one answer fits

Answer: B