The function is given by f(x) = (4)/(x-3) + 4. Calculate the coordinates of M, D and the value of t if y=x+t is a line of symmetry.

Here are the solutions to the questions: 4.1 Write down the coordinates of M. The function is given by f(x) = (4)/(x-3) + 4. For a function of the form y = (a)/(x-p) + q, the vertical asymptote is x=p and the horizontal asymptote is y=q. The point of intersection of the asymptotes is (p, q). In this case, p=3 and q=4. So, the vertical asymptote is x=3 and the horizontal asymptote is y=4. The point M is the intersection of these asymptotes. M = (3, 4) The coordinates of M are (3, 4). 4.2 Calculate the coordinates of D. D is the y-intercept of f(x). To find the y-intercept, set x=0. f(0) = (4)/(0-3) + 4 f(0) = (4)/(-3) + 4 f(0) = -(4)/(3) + (12)/(3) f(0) = (8)/(3) The coordinates of D are (0, (8)/(3)). 4.3 If y=x+t is the equation of a line of symmetry of f, calculate the value of t. The lines of symmetry for a hyperbola of the form y = (a)/(x-p) + q pass through the point of intersection of the asymptotes (p, q) and have gradients 1 and -1. From 4.1, the point of intersection of the asymptotes is M (3, 4). The given line of symmetry is y=x+t, which has a gradient of 1. Substitute the coordinates of M (3, 4) into the equation y=x+t: 4 = 3 + t t = 4 - 3 t = 1 The value of t is 1. 4.4 Determine the values of x for which f(x) 0. We need to solve the inequality (4)/(x-3) + 4 0. (4)/(x-3) -4 Divide by 4: (1)/(x-3) -1 Consider two cases for x-3: Case 1: x-3 > 0 x > 3 Multiply both sides by (x-3) (the inequality sign does not change): 1 -1(x-3) 1 -x + 3 x 3 - 1 x 2 This contradicts our assumption x > 3. So, there are no solutions in this case. Case 2: x-3 < 0 x < 3 Multiply both sides by (x-3) (the inequality sign reverses): 1 -1(x-3) 1 -x + 3 x 3 - 1 x 2 Combining with our assumption x < 3, we get 2 x < 3. The values of x for which f(x) 0 are 2 x < 3. 4.5 Calculate the coordinates of A. A is the point on f closest to M. These points lie on the lines of symmetry. From the graph, A is in the first quadrant, so it lies on the line of symmetry with a positive gradient, which is y=x+1 (from 4.3). To find the coordinates of A, we set f(x) = x+1: (4)/(x-3) + 4 = x+1 (4)/(x-3) = x+1-4 (4)/(x-3) = x-3 Multiply both sides by (x-3): 4 = (x-3)^2 Take the square root of both sides: ±sqrt(4) = x-3 ± 2 = x-3 This gives two possible values for x: 1. 2 = x-3 x = 5 2. -2 = x-3 x = 1 If x=5, then y = 5+1 = 6. So, (5, 6). If x=1, then y = 1+1 = 2. So, (1, 2). From the graph,