Question

1.Let functions f and g be defined by f(x)=2x+1, and
g(x)=x^2-2, respectively. Find
a)(gof)(a+2)
b)(fog)(a+2)

2.Let A={x:x≠2) and define f: A→R by f(x)=4x/(2x-1) . Is f is one- to- one?
Find the range of f . Then find f^(-1) and hence determine the domain and range of f^(-1)

Answer 1

(g◦f) = g(f) = f^2-2 = (2x+1)^2-2
(f◦g) = f(g) = 2g+1 = 2(x^2-2)+1

x=4f/(2f-1)
2xf-x = 4f
f(2x-4) = x
f = x/(2x-4)

So, the range of f is all reals except x=2. The domain is all reals except 2 and 1/2.

The domain of f^-1 is the range of f. The range is all reals except x=1/2.