Absolute values and the correct inverse branch

For real x≠−1/2x\ne-1/2, f(x)=∣(1−2x)/(2x+1)∣f(x)=|(1-2x)/(2x+1)|; for all real x, g(x)=tan⁡−1(2x)g(x)=\tan^{-1}(2x).

(i) Show f is not one-one. Find the least k such that restricting the domain to [k,∞)[k,\infty) makes it one-one. marks On this restricted domain, (ii) define the inverse of f. marks

(iii) Find the exact range of gf−1gf^{-1}. marks

[2011/AJC/Promo/8]

Total marks: 7

Solution

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

  1. (i) One-one restrictions →
  2. (ii) Inverse of a restricted absolute value →
  3. (iii) Range of a composite →