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If f(x)=(1)/(2)x-4 and f(g(x))=g(f(x)), ...

If `f(x)=(1)/(2)x-4` and `f(g(x))=g(f(x))`, which of the following can be g(x) ?
I. `2x-(1)/(4)`
II. `2x+8`
III. `(1)/(2)x-4`

A

I only

B

II only

C

III only

D

II and III only

Text Solution

Verified by Experts

The correct Answer is:
D
If `f(g(x))=g(f(x))`, then it doesn't matter which function you apply first , you should get the same result. This will happen under two sets of circumstances. The first is when the functions are inverses, in which case both `f(g(x))` and `g(f(x))` will get you back to the x you started with. The second is when the two functions are identical, in which case `f(g(x))` and `g(f(x))` are identical. II is the inverse of f(x) and III is the identical function f(x), so II and III are possible.
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