Restrict the domain to choose an inverse

For all real x, f(x)=2/[1+(5x−1)2]f(x)=2/[1+(5x-1)^2].

(i) Find its range. marks

(ii) Explain why it has no inverse. mark

(iii) Restrict its domain to x≥kx\ge k: find the least k that gives an inverse, and find that inverse. marks

(iv) Sketch the restricted f and its inverse together, clearly showing their relationship. marks

(v) Show algebraically that a solution of f(x)=f−1(x)f(x)=f^{-1}(x) satisfies 25x3−10x2+2x−2=025x^3-10x^2+2x-2=0. marks

[2011/IJC/Prelim/I/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) Range →
  2. (ii) One-one functions →
  3. (iii) Restricting a domain →
  4. (iv) Graph of an inverse →
  5. (v) Fixed points →