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Question

A function f is defined by f : x -> (x+1)/(x-1), x ≠ 1. If f ^ 2 * (2) = 3f ^ - 1 *(a) , find the value of a.

Answer 1

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

is graphically a rectangular hyperbola

which is symmetric about the point (1,2)

with asymptotes x=1 and y=2

to find the inverse switch x and y then solve for the new y

x=(y+1)/(y-1)

f^-1(x)=y=(x+1)/(x-1)=f(x)

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

if (x,y) is on f(x) then (y,x) is on f^-1(x)

2f^2(2) = 3f^-1(a)=3f(a)

2[(2+1)/(2-1)]^2= 3(a+1)/(a-1)

2(9)=3(a+1)/(a-1)

6=(a+1)/(a-1)

6a-6=a+1

6a-a=1+6

a=7