Question 1186585
Given that the (restricted) domain of f(x) = 3x - 2 is [0,{{{infinity}}}), we need  to choose the values of {{{g(x) = 2x^2 - 8 >= 0}}} for the composition f o g to be possible.


(a)  Hence, {{{2x^2 - 8 >= 0}}} ===> {{{x^2 >= 4}}}, or x ∈ ({{{-infinity}}}, -2] U [2, {{{infinity}}}).  By virtue of continuity, the greatest value of k for which the composite function f o g can be formed is {{{red(-2)}}}.


(b)  (i)  Now {{{(f o g)(x) = f(g(x)) = 3(2x^2 - 8)-2 = 6x^2  -26}}}.  Since the domain of f o g is ({{{-infinity}}}, -3], the expression {{{6x^2  -26}}} maps ({{{-infinity}}}, -3] onto the interval [28, {{{infinity}}}).  
This is the range of (f o g)(x).


(ii)  To find {{{(f o g)^(-1)(x)}}}, let {{{y =  6x^2  -26}}}.

===> {{{x^2 = (y+26)/6}}}  ===> {{{x = -sqrt((y+26)/6)}}}.  (Choose the negative part since this is an element of ({{{-infinity}}}, -3].)

===>  {{{y = -sqrt((x+26)/6)}}} after interchanging the places of x and y.


===>  {{{(f o g)^(-1)(x) = -sqrt((x+26)/6)}}}.  Its domain is the range of f o g, namely [28, {{{infinity}}}), while its range is the domain of f o g, namely ({{{-infinity}}}, -3].