Q25 of 45 Page 1

Consider f: given by Show that f is bijective. Find the inverse of f and hence find f-1(0) and x such that f-1(x) = 2.

Given function is



To prove f is bijective


one-one


Let x1, x2∈


d 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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