Recover the unknown function by undoing a logarithm

Functions f:x↦1+2x−1 f:x\mapsto1+\frac2{x-1}, x∈Rx\in\mathbb R, x<1 x<1; g:x↦ln⁡xg:x\mapsto\ln x, x∈Rx\in\mathbb R, 0<x<10<x<1.

(i) Explain why composite gfgf does not exist. mark

(ii) Find an expression for fg(x) fg(x). Hence or otherwise find (fg)−1(0)(fg)^{-1}(0). marks

(iii) Find h(x) h(x) for each case: (a) gh(x)=xgh(x)=x, mark (b) hg(x)=x2+1 hg(x)=x^2+1. marks

[2018/DHS/Prelim/I/5]

Total marks: 8

Solution

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

  1. (i) Existence of composites →
  2. (ii) composite Defining composites →
  3. (ii) inverse value Inverse values →
  4. (iii)(a) Recovering a function →
  5. (iii)(b) Recovering a function →