2.5: Rational derivative and linear composite

0606/22/M/J/17 — Question 12 · 7 marks

The function gg is defined, for x>−12x > -\tfrac{1}{2}, by g(x)=32x+1g(x) = \dfrac{3}{2x + 1}.
(i)
Show that g′(x)g'(x) is always negative.
[2]
(ii)
Write down the range of gg.
[1]
(iii)
The function hh is defined, for all real xx, by h(x)=kx+3h(x) = kx + 3, where kk is a constant.
Find an expression for hg(x)hg(x).
[1]
(iv)
Given that hg(0)=5hg(0) = 5, find the value of kk.
[2]
(v)
State the domain of hghg.
[1]