hi please explain every question in depth i don't only need the answer i a need a step-by-step solution so i can understand please show all your work!

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Unit 6 Activity 2 - Summative MHF4U - DCE eLearning Background Patient #: 11235813 Surname: Mathews Given Name: Sten Ulysses DOB: June 6th, 1995 Height: 182.9 cm Weight: 61.4 kg Location: Welland, Ontario On December 29th, 2020, Mr. Mathews was examined by Dr. Andersen at the General Hospital in Welland, Ontario. Mathews complained of chronic excess gas, abdominal bloating, distension, diarrhea, and abdominal pain. The patient reported that his symptoms have been re- occurring and have fluctuated in intensity over the past eighteen months. Mathews initially theorized that this condition was the result of a poor diet, consisting mainly of greasy "fast" foods. Over the last two months Mathews had changed his eating habits and lifestyle to include healthy foods and exercise. This modification did not have any effect on his condition, and he was concerned about his dramatic weight loss over the past three months. Mathews appeared distraught and genuinely concerned for his health. Polynomial Functions: 10 marks Rational Functions: 10 marks Trigonometric Functions: 10 marks Exponential Functions: 6 marks Logarithmic Functions: 8 marks Characteristics of Functions: 6 marks Total Marks: 50 marks 1Unit 6 Activity 2 - Summative MHF4U - DCE eLearning Part A: The Anatomy Dr. Andersen, a specialist on the human gastronomic system, determined that many of the symptoms elicited by Mathews could be directly related to a problem in either the small or large intestine. A battery of tests were performed on Mathews, two noteworthy results are described below. The first procedure was performed in the interest of collecting bacterial culture swabs of Mathews' small intestine. A long flexible tube is passed through the nose, down the throat and esophagus and through the stomach. A small camera, attached to the top of the tube recorded every twist and turn of the journey. It was performed under x-ray guidance. The data from both the camera and the x-ray machine were used to create a detailed sketch of Mathews' gastronomic tract. Question 1 (10 marks) A specific section of Mathews' gastronomic tract can be modeled by the function g(x)=-x"+11x" 43x" +69x\" 36x, where x represents distance traveled by the scope, in cm, and g(x) refers to the vertical height within the body relative to the belly button, in cm. a) Rewrite this equation in factored form. Show all your work. (5 marks) b) Use this information to sketch a graph, by hand, of this section of Mathews' small intestine. (2 marks) c) Determine the domain of this function. (1 mark) d) Bacterial culture samples were taken at two unique points along the journey. Clearly mark these points on your graph. (2 marks) e At the first turning point e At the only root with order two The second procedure was performed in the interest of collecting a biopsy of a section of the small intestine. The biopsy was removed surgically and examined quantitatively and qualitatively. Faint, yet distinct, areas of light and dark tissue were discovered on the surface of the small sample. Unit 6 Activity 2 - Summative MHF4U - DCE eLearning Assume that a 20 mm by 20 mm section of tissue was placed on tops of a Cartesian plane with is centre at the origin. The darker, shaded areas found on the tissue sample satisfy the : ; : ; x+1 following rational inequality: T >0 22 a) Sketch a graph, by hand, of the function F(x)=2*L (8 marks) x-9 A f = = \\ 3 Ry b) Shade the region(s) of the graph that satisfy the b inequality to reveal the light and dark regions f of the biopsy noted by Dr. Andersen. Part B: The Diagnosis With the help of Mathews, Dr. Andersen constructed a timeline of the last 18 months. They mapped regions based on the intensity of his symptoms and found that there was a cyclical pattern. Question 3 (6 marks) The cyclical pattern of symptoms is sinusoidal in nature. When graphed as a function of intensity, g(x), and time, x in months, the function was very similar to f(x} =sinx. Dr. Andersen described the shape of this graph, with reference to f(x) =sinx. Intensity is ranked on a scale of 0 to 10, where zero represents normal and ten represents filll blown symptoms. The amplitude of the cyclical pattern is 5 and the equation of the axis of the curve is y =5. The period of the function is 4 months. Using f (x)=sinx as a base function, the symptoms' finction has been horizontally translated 2 months to the right. a) Write a possible equation for the function g (x) (4 marks) b) Sketch a graph of the function, g(x), by hand on the interval 0 5log(x+1)+10models the concentration of levofloxacin in mol/L over time x, in hours. The function M[x) = llc-g(x +1}+5m0dels the concentration of metronidazole in mol/L over time x, in hours. a) Which of the two drugs has a higher initial concentration in the blood stream? Justify your answer with an explanation. (2 marks) b) Algebraically determine when L(x)=M (x) and state what this represents in this situation. (6 marks) c) If Mathews is instructed to take both antibiotics at once, the concentration levels could be modelled by the function (L+M )(x). How would the graph (L+M )(x) differ from the individual graphs of L (x) and M (x)? Explain your answer. (2 marks) After a short period of time, Mathews made a full recovery. However, due to his below average gastrointestinal muscular activity, he must continue the treatment indefinitely to avoid further overgrowth