Inverse function

Question

Inverse function

Given f(x) = ax+b / cx+d, find f^-1(x).

(a) Using the formula f^-1 obtained, determine f^-1 for each of the following functions.

(i) f(x) = x+8 / x-5 , x

(ii) f(x) = 2x-3 / x+4 , x -4

(b) If c 0, what are the conditions on a,b,c and d so that f = f^-1?

Answer

(a) (i) f^-1(x) = 8+5x / x-1 , x 1

(ii) f^-1(x) = -3-4x / x-2 , x 2

(b) f = f^-1 if a = -d

Can give me the clear solution of the question? Thank you

Answer 1

f(x) = (x+8)/(x-5) =y

switch x and y

(y+8)(y-5) =x

solve for the new y

y+8 = xy -5x

y-xy = -5x-8

y = -(5x+8)/(1-x) = f^-1(x) = inverse function

f^-1(x) = (8+5x)/(x-1)

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

the composite of the inverse functions = x

y=(2x-3)/(x+4)

switch

x = (2y-3)/(y+4)

solve for y

2y-3 = xy+4x

2y-xy = 4x+3

y = (4x+3)/(2-x) = f^-1(x) = the inverse function

(4x+3)/(2-x) = (-3-4x)/(x-2) after multiplying both numerator & denominator by -1

f(x) = (x+a)/(x+b)

f^-1(x) = (x-d)/(x+c)

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

when a=-d

and b=c