​ Let f(x) = x-1/x-3 and g(x) = x-3/x-1 , x≠1 be two functions defined on R. (i) Find ƒog(x), x ≠ 0 ​

Question

Let f(x) = \(\frac{x-1}{x-3}, x≠3\)and g(x) = \(\frac{x-3}{x-1}\), x≠1 be two functions defined on R.

(i) Find ƒog(x), x ≠ 0

(ii) Find ƒ^-1 (x) and g^-1 (x), x ≠ 1

(iii) Find (goƒ)^-1 (x)

Answer 1

(ii) f(x) = \(\frac{x-1}{x-3}\) = y ⇒ x-1 = y (x-3)

⇒x-1 = xy -3y ⇒ x - xy = 1-3y

⇒ x(1-y) =1-3y ⇒ x = \(\frac{1-3y}{1-y}\)

Interchanging x and y we have;

g^-1 (x) = \(\frac{x-3}{x-1}\)

(iii) We have ; (g of)^-1 (x) = f^-1 og^-1 (x)