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

If this part uses an earlier result, revisit (i) Existence of composites or open all working together.

(ii) composite

Defining composites

Find an expression for fg(x)fg(x).

Hint

  1. Substitute g(x)=ln⁡xg(x)=\ln x into ff and combine the fractions.

Answer · (ii) composite

fg(x)=(ln⁡x+1)/(ln⁡x−1)fg(x)=(\ln x+1)/(\ln x-1), 0<x<10<x<1.