Question

28.Statement I A function ∶ → satisfies the equation f(x) – f(y) = x – y, ∀ x, y ∈ R and
f(x) = 2, then f(xy) = xy – 1
Statement II = (1/x) , ∀ ∈ , ≠ 0 and (2 )=7/3
() =(^2+ + 1)/(^2− + 1)
(a) Statement I is true, Statement II is true; Statement II is a correct explanation for
statement I
(b) Statement I is true, Statement II is true; Statement II is not a correct explanation for
statement I
(c) Statement I is true, Statement II is false
(d) Statement I is false, Statement II is true

Answer 1

Answer:

f:R→R

f(x)f(y)−f(xy)=x+y∀x,yϵR

∵f(1)>0

Put,x=1&y=1

⇒f(1)f(1)−f(1)=2

⇒f

2

(1)−f(1)=0

f

2

(1)−2f(1)+f(1)−2=0

f(1)(f(1)−2)+1(f(1)−2)=0

(f(1)+1)(f(1)−2)=0

f(1)=−1&f(1)=2

∴f(1)=2True.

Put,y=1

2f(x)−f(x)=2+x

⇒f(x)=2+x

⇒y=x+2

x=y−2

f

−1

(x)=x−2

⇒f(x)f

−1

(x)=(x+2)(x−2)

f(x)f

−1

(x)=x

2

−4