Merging similar topics. Please refer to the solution above and ask if anything is unclear.

Similar questions to practice:
for-which-of-the-following-functions-is-f-a-b-f-b-f-a-124491.html
for-which-of-the-following-functions-f-is-f-x-f-1-x-for-85751.html
for-which-of-the-following-functions-is-f-a-b-f-a-f-b-for-93184.html
let-the-function-g-a-b-f-a-f-b-143311.html

Hope this helps.
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Hi,

To solve this easiest way is just put the value and see that if it equals or not.

with option 1. f(a) = a^2 and f(b) = b^2

so L.H.S = a^2 - b^2

and R.H.S = (a-b)^2 ==> a^2 + b^2 -2ab.

so L.H.S not equal to R.H.S


with option 2. f(a) = a/2 and f(b) = b/2

L.H.S = a/2 - b/2 ==> 1/2(a-b)

R.H.S = (a-b)/2

so L.H.S = R.H.S which is the correct answer.

you can try simile for other option.

Thanks,
AB

+1 Kudos if you like and understand.

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

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|

My approach:
one can be easily confused in this type of question.. so please proceed slowly.

1. f(x)= x^2
LHS --> a^2 - b^2
RHS --> (a-b)^2

LHS != RHS ---> Reject

2. f(x) = x/2
LHS --> (a/2) - (b/2)
RHS --> (a-b)/2

LHS = RHS

No need to check further

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