If f : R - {- 4/3} → R be a function defined as f (x) = 4x/(3x + 4). The inverse of f is the map
g : Range f → R - {- 4/3} given by :
A. g (y) = 3y/(3 - 4 y)
B. g (y) = 4y/(4 - 3y)
C. g (y) = 4y/(3 - 4 y)
D. g (y) = 3y/(4 - 3y)

Solution:

It is given that f : R - {- 4/3} → R is defined as f (x) = 4x/(3x + 4)

Let y be an arbitrary element of Range f

Then, there exists x ∈ R - {- 4/3}

such that y = f (x).

⇒ y = 4x / (3x + 4)

⇒ 3xy + 4 y = 4x

⇒ x (4 - 3y ) = 4 y

⇒ x = 4y/(4 - 3y)

Define f : R - {- 4/3} → R as

g (y) = 4y/(4 - 3y)

Now,

(gof)(x) = g (f (x)) = g (4x/(3x + 4))

= {[4 (4x/(3x + 4))] / [4 - 3 (4x/(3x + 4))]}

= 16x/(12x + 16 - 12x)

= 16x/16 = x

And

(fog)(x) = (g (x)) = f (4y/(4 - 3y))

= {[4 (4y/(4 - 3y))] / [3 (4x/(3x + 4)) + 4]}

= 16y/(12y + 16 - 12y)

= 16y/16 = y

⇒ gof = I_{R - {- 4/3}} and fog = I_{Range f}

Thus, g is the inverse of f i.e., f ^-1 = g

Hence, the inverse of f is the map g : Range f → R - {- 4}, which is given by

g (y) = 4y/(4 - 3y)

The correct answer is B

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.3 Question 14

If f : R - {- 4/3} → R be a function defined as f (x) = 4x/(3x + 4). The inverse of f is the map g : Range f → R - {- 4/3} given by : A. g (y) = 3y/(3 - 4 y) B. g (y) = 4y/(4 - 3y) C. g (y) = 4y/(3 - 4 y) D. g (y) = 3y/(4 - 3y)

Summary:

For the function f : R - {- 4/3} → R be a function defined as f (x) = 4x/(3x + 4).g is the inverse of f i.e., f ^-1 = g. Hence, the inverse of f is the map g : Range f → R - {- 4}, which is given by g (y) = 4y/(4 - 3y) .The correct answer is B