Composition conditions and a hidden constant case

Functions f:x↦(x2−6x−6)/(x+1) f:x\mapsto(x^2-6x-6)/(x+1), x∈Rx\in\mathbb R, x≠−1 x\ne-1; g:x↦(ax+1)/(x+b) g:x\mapsto(ax+1)/(x+b), x∈Rx\in\mathbb R, x≠−bx\ne-b, where a,ba, b are constants.

(i) Sketch y=f(x) y=f(x), giving coordinates of turning points and equations of asymptotes. Write down the range of ff. marks

(ii) Find the value of aa and range of bb such that both fgfg and gfgf exist. marks

(iii) Find g−1(x) g^{-1}(x). Given g−1(x)=g(x) g^{-1}(x)=g(x) for all real xx, x≠−bx\ne-b, find bb in terms of aa. marks

[2019/TJC/Promo/9]

Total marks: 10

Solution

Choose a part to attempt. Each page keeps the full question available and loads only that part’s working and exploration.

  1. (i) Sketching rational functions →
  2. (ii) Existence of composites →
  3. (iii) Self-inverse functions →