(1) Define the function f : R \\ {0} - R by f (x) = ? 4x (a) Sketch the graph of y = f(x). (b) Find the range of f. (c) Determine whether or not f is bijective. (d) Define a function g : 1\\ {0, ?} - R by g(x) = fof(x). Show that g can be written in the form g(x) = ax+b where a and b are constants. (e) Show for any x E R \\ {0, }} that fo fo f (x) = x. [2+2+2+2+2 =10 marks ] (2) In 2021, detailed collaborative archaeological work was published regarding the use of Cloggs Cave, near Buchan in East Gippsland, Victoria. (If you're interested, you can read about it in The Conversation or the research article. Links embedded in underlined text.) One of the findings (simplified) was that the Main Chamber of the cave contained human artefacts at least 19,000 years old.(4) Determine the following limits, justifying your answers carefully. IN 1 (a) lim x-1 X VxVx+3 cost(x) - 1 (b) lim x-+0 x-1 (c) lim x-1x3 - 1 x- +5x +3 (d) lim x-co x(4x+7) (e) lim v x sin X-+ 0 X 3 [2+2+2+2+2 =10 marks ] (5) Let f and g be continuous functions defined on R and suppose that f (2) = 2 and x- 2 lim (f (x) - 5f (x) g(x)) = -28. (a) Determine the value of g (x) for x = 2. (b) Calculate the limit lim x sin x - 2 f ( x ) + g(x) cos(x - 2) . [ 4+4=8 marks ]You may assume that, in living organisms, carbon-14 is constantly replenished, but after an organism dies, the carbon-14 decays with a half-life of 5,730 years. For or- ganic material associated with artefacts 19,000 years old, what fraction of carbon-14 atoms would remain? Give your answer to 3 significant figures. (You may use a com- puter/calculator/etc for the calculation but show all working.) [5 marks ] (3) For a real constant a E R, define the piecewise function h : R - R by xta ifx