Let fz 5x4x 7 This function is onetoone Find the inverse f

Let f(z) = -5x/4x - 7. This function is one-to-one. Find the inverse function f^-1(x), and verify that it is the inverse by explicitly computing (f f^-1)(x) and (f^-1 f) (x).

Solution

f(x) = -5x/(4x -7)

y = -5x/(4x -7)

To find inverse of f(x) , replace x = y and y = x . Solve for y:

x =-5y/(4y -7)

4xy - 7x = -5y

(4x +5)y = 7x

y = 7x/(4x +5)

f^-1(x) = 7x/( 4x +5)

(fo f^-1)(x) : -5(7x/(4x +5) /(4*7x/(4x +5) -7)

= -35x/( 28x -28x -35)

= x

-5x/(4x -7)

(f^-1 o f)(x) : 7(-5x/(4x -7) /( 4(-5x/(4x -7) +5)

= -35x/( -20x +20x -35)

= x

Hence (f o f^-1)(x) = ( f^-1 o f)(x) = x