Q. 25 A5.0( 3 Votes )

# Consider f: <span

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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