MAT1510 Assignment 2 (COMPLETE ANSWERS) 2024 (186115) - DUE 31 May 2024

MAT1510 Assignment 2 (COMPLETE ANSWERS) 2024 (186115) - DUE 31 May 2024...Detailed work, solutions, memos, notes, and explanations.... ASSIGNMENT 02 Due date: Friday, 31 May 2024 Total Marks: 100 UNIQUE ASSIGNMENT NUMBER: 186115 ONLY FOR YEAR MODULE This assignment covers chapter 2 of the prescribed book as well as the study guide DO NOT USE A CALCULATOR. Question 1: 13 Marks Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. (1.1) If f is a function, then f(s + t) = f(s) + f(t). (3) (1.2) If f(s) = f(t), then s = t. (2) (1.3) If f is a function, then f(3x) = 3f(x). (3) (1.4) A vertical line intersects the graph of a function at most once. (2) (1.5) If f is one-to-one, then f (3) − 1 (x) = 1 f(x) . Question 2: 9 Marks The perimeter of a rectangle is 16 meters. (2.1) If the length of one of the sides of the rectangle is (1 + x) meters, express the area A of the (4) rectangle in terms of x. (2.2) Calculate the maximum area of the rectangle. (3) (2.3) What are the dimensions of the rectangle when its area is a maximum? (2) Question 3: 6 Marks Suppose a stone is thrown vertically upwards with a velocity ofu meters per second. Then its height is h (in meters) after t seconds is given by the formula h = ut − 4.8t 2 . (3.1) Suppose the stone is thrown upwards with a velocity of 24 meters per second. Sketch the (5) graph of the function defined by h = ut − 4.8t 2 . Label the axes properly, and show the coordinates of the critical points on the graph clearly. 16 MAT1510/101/0/2023 (3.2) What is the maximum height that the stone reaches? (1) Question 4: 6 Marks Sketch the graph of the function g(x) which is piecewise-defined by (4.1) (4) g(x) =    − x 2 + 2x + 3 if x < 1 4 if x = 1 x 2 − 2x + 5 if x > 1 (4.2) Explain why g is called a function. (1) (4.3) Is g a one-to-one function? Give a reason for your answer. (1) Question 5: 20 Marks The sketch shows the graph of the functions f and g. Function f is defined by y = f(x) = m|x − p| + q 17 and g is defined by y = g(x) = ax2 + bx + c S is the salient point of the graph of f , and T is the turning point of the graph of g. The two graphs intersect each other at T and R. (5.1) Determine the value of m, p and q and then write down the equation of f. (4) (5.2) Describe the steps of the transformation process that you would apply to the graph of f to (3) obtain the graph of y = 5|x|. (5.3) Find the values of a, b and c and then write down the equation of g. (4) (5.4) Determine the coordinates of T. (2) (5.5) Use the graph of f and g to solve the inequality (3) f(x) g(x) < 1 for x ∈ (−2,4). (Do not solve the inequality algebraically.) (5.6) Suppose the function d describes the vertical distance between the graph of f and g on the interval [− 2, xT ] ( where xT is the x-coordinate of T). (a) Complete and simplify the equation (2) d(x) = ................ for x ∈ [−2, xT ]. (b) What is the maximum vertical distance between the graphs of f and g on the interval (2) [− 2, xT ]. Question 6: 11 Marks Suppose a function g is defined by y = g(x) = 7+ 6x − x 2 (6.1) Restrict the domain of g such that the function gr defined by (2) gr (x) = g (x) for all x ∈ Dgr , is one-to-one function, and such that the domain Dgr contains only positive numbers. (6.2) Determine the equation of the inverse function g (3) − 1 r and the set Dg − 1 r . (6.3) Show that (6) gr ◦ g − 1 r (x) = x for x ∈ Dg − 1 r and g − 1 r ◦ gr (x) = x for x ∈ Dgr . 18 MAT1510/101/0/2023 Question 7: 15 Marks The function h(x) and l(x) are defined by h(x) = 3− √ x + 4 and l(x) = 3+ √ x + 4. Suppose the symbols Dh and Dl denote the domains of h and l respectively. Determine and simplify the equation that defines (7.1) the domain of h and l respectively, Dh and D (2) l (7.2) h + l and give the set Dh+ (3) l (7.3) h − l and give the set Dh− (3) l (7.4) h ·l and give the set Dh· (3) l (7.5) (4) h l and give the set Dh l . Question 8: 16 Marks The functions f and g are defined by f(x) = x x + 1 and g(x) = x + 1 x respectively. Suppose the symbols Df and Dg denote the domains of f and g respectively. Determine and simplify the equation that defines (8.1) f ◦ g and give the set D (3) f◦g (8.2) g ◦ f and give the set Dg◦ (3) f (8.3) f ◦ f and give the set D (3) f◦f (8.4) g ◦ g and give the set Dg◦ g (3) (8.5) Find any two possible functions h and l such that (4) (h ◦ l) (x) = − √ x + 2 3 . Question 9: 4 Marks (9.1) Stretch the graph of f(x) = | x| horizontally by a factor 2, then shift the resulting graph hori- (2) zontally 6 units to the left, and then shift this transformed graph vertically to 4 units downward. What is the equation for the final transformed graphs? (9.2) Shrink the graph of g(x) = x (2) 2 vertically by a factor of 1 6 , then reflect the resulting graph about the x-axis, and then shift this transformed graph vertically 5 6 units upwards. What is the equation for the final transformed graph?