Q. 25 A5.0( 3 Votes )

Consider f: <span

Answer :

Given function is



To prove f is bijective

We need to prove that f is one-one and onto.

one-one

To Prove one-one,
f(x1) = f(x2)
And, x1 = x2

Therefore,

Let x1, x2


f(x1) = f(x2)



12x1x2 + 9x2 + 16x1 + 12 = 12x1x2 + 9x1 + 16x2 + 12


7x1 = 7x2


x1 = x2


Hence f is one-one.


Onto


Let y


Then,


f(x) = y






Clearly, x R for all y . Also, x ≠ (-4/3).


Because, x = (-4/3)



12y - 9 = -16 + 12y


-9 = -16 (which is not possible)


Thus, for each y there exists


such that



So, f is onto.


Thus, f is both one-one and onto. Consequently, it is invertible.


Now,


f(f-1(x)) = x for all x R - [4/3]






Now,



Also given that-




4x-3 = 8-6x


10x = 11


x = 11/10

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