The function f(x) is defined as f(x) = 1/(x + 1), where x is not equal

Question

The function f(x) is defined as  f(x)=\frac{1}{x+1} , where x is not equal to -1. If a is not equal to -1 or 0, which of the following expressions must be true for all values of a?

(I) f(a − 1) = f(a) – f(1)

(II)  \frac{2}{f(a−1)}=\frac{2}{f(a)}−\frac{1}{f(1)}

(III) f(a − 1) = f(a)∗f(1)

(A) I only
(B) II only
(C) III only
(D) I, II and III
(E) None of the above

chetan2u · Answer

Let us find value of f(a), f(1) and f(a-1) if a=3
So 
1)  f(1) = \frac{1}{(1+1)}=\frac{1}{2} 
2)  f(a)=f(3)=\frac{1}{(3+1)}=\frac{1}{4} 
3)  f(a-1)=f(3-1)=f(2)=\frac{1}{(2+1)}=\frac{1}{3}

Let us see the choices

(I) f(a−1)=f(a)–f(1)
So  \frac{1}{3}=\frac{(1}{4})-\frac{1}{2} ....NO

(II) \frac{2}{f(a−1)}=\frac{2}{f(a)}−\frac{1}{f(1)} 
 2/(\frac{1}{3})=2/(\frac{1}{4})-1/(\frac{1}{2}).....2*3=2*4-1*2.....6=8-2 ... YES

(III)f(a−1)=f(a)∗f(1)
 \frac{1}{3}=\frac{1}{4}*1\frac{}{2}. ....NO

Only II

B

The function f(x) is defined as f(x) = 1/(x + 1), where x is not equal

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