MAT1510/101 ASSIGNMENT 03 Due Date: 17 June 2016 Total Marks: 100 UNIQUE ASSIGNMENT NUMBER: 736826 ONLY FOR YEAR MODULE STUDY MATERIAL: SRW and STUDY GUIDE: CHAPTER 4 The answers to only some of the questions will be marked. DO NOT USE A CALCULATOR. Question 1: 16 Marks (1.1) Sketch the graph of the function g defined by (4) y = 23x 4. State the equation of the asymptote, the domain, and the range of the function g. The graph does not need to be accurate, but show clearly where (if) the graph cuts the axes. (1.2) Suppose the function f is defined by (2) y = 2x . Explain all the steps of the transformation process that you would apply to the graph of g to obtain the graph of f . (1.3) Determine an equation for the inverse function g 1 . (4) (1.4) \u0001 \u0001 Show that g g 1 (x) = x and g 1 g (x) = x. (6) Question 2: 16 Marks The sketch shows the graphs of the functions f and g where 2 y = f (x) = |x 1| + 2 3 and y = g(x) = loga (x + k ), where a, k R; a > 0 and a 6= 1. D is a point on the graph of g, C = (5, 0) and DC is parallel to the yaxis. 25 y y = g(x) D A B x C y = f (x) (2.1) Determine the coordinates of A and B. (4) (2.2) Use the graphs to determine a and k and hence the equation of the function g. (5) (2.3) Use the graphs (not the algebraic expressions for f (x) and g(x)) to solve the inequality g(x) > f (x). (3) (2.4) Give the equation of the asymptote of the graph of g. (1) (2.5) Determine the length of the line segment DC. (3) Question 3: 10 Marks (3.1) Rewrite the expression (4) q 3 2 (x + 1) log5 3 3x + 7 in a form with no logarithms of products, quotients, or powers. (3.2) Rewrite the expression (4) ln (x 3) + ln (x + 3) \u0001 1 ln x 2 + 9 2 as a single logarithm. (3.3) 26 Explain why log5 x 1 6= log5 x \u00011 . (2) MAT1510/101 Question 4: 15 Marks Suppose the functions f , g and h are defined by f (x) = 22x+1 + 2x+2 , and g(x) = 3 2x , and h(x) = log4 (x 1) log4 (x + 2)1 . (4.1) Write down the sets Df , Dg and Dh that represent the domains of f , g and h respectively. (3) (4.2) Solve the equation f (x) = g(x) for x. (6) (4.3) Solve the inequality h(x) 1 for x. (6) Question 5: 8 Marks Suppose the functions l and k are defined by l(x) = log2 x 4 k(x) = log4 x 8 and Find the coordinates of the point(s) of intersection of the graphs of l and k with each other. Question 6: 8 Marks y f 0 x (2, 3) 27 (6.1) The graph of the function f is given in the sketch. Use the graph to find the equation of the form f (x) = ax+2 + b which defines the function f (i.e. determine the values of a and b). (6) (6.2) What are the steps of transformation that you would apply to the graph of f to obtain the graph of g where g(x) = ax ? (2) Question 7: 8 Marks y 0 x 3 (12, 2) (7.1) \u0010x \u0011 . Use the c graph to determine c and p and hence write down the equation that defines h. (6) What is the equation of the graph that is obtained by reflecting the graph of h in the line y = x? DO NOT USE A CALCULATOR WHEN YOU ANSWER THE FOLLOWING QUESTIONS. (2) The sketch shows the graph of the function h defined by h(x) = logp (7.2) Question 8: 6 Marks The number of bacteria in a culture is given by the formula n(t) = 500e0.5t where t represents time, measured in hours. (8.1) What is the relative rate of growth of this bacterium population? Express your answer as a percentage. (1) (8.2) What is the initial bacterium population in the culture (at t = 0)? (1) (8.3) After how many hours will there be 1500 bacteria? Leave your answer in terms of ln. (4) 28 MAT1510/101 Question 9: 8 Marks Suppose 2 kg of a radioactive substance decays to 250 g in 75 years. What is the half-life of the substance? Question 10: 5 Marks The intensity I (measured in watts/m2 ) of a sound system at a rock concert is measured at 1 watt/m2 1 watt/m2 on the second night, what difference will it make on the first night. If it is measured at 100 to the intensity level (measured in decibels)? 29