Provide references so that i will be able to follow.
4.2. EXERCISES 13 (14) Let f(r) = Vr, g(x) = , and h(x) = r'. Find (ho ((hogof) - f))(4). Answer: (15) Let f(x) = r+ 7, g(r) = vr + 2, and h(r) = 12. Find (ho((fog) - (9. f)))(7). Answer: (16) Let f(x) = v5 -r, g(x) = vr+ 1 1, h(x) = 2(x - 1)-1, and j(r) = 4r -1. Then (f o (g + (hog)(ho )))) (5) = (17) Let f(r) = x', g(x) = v9 + r, and h(r) = =_ . Then (ho(fog-90 /))(4) = (18) Let f(x) = r', g(x) = v9 + r, and h(r) = (r- 1)13. Then (ho ((fog)(go f)))(4) = (19) Let f(r) = =, g(x) = vr, and h(r) = r + 1. Then (g(fog) + (go f o h))(4) = (20) Let g(x) = 5 - r', h(r) = vr+ 13, and j(r) = =. Then (johog)(3) = (21) Let h(x) = = Vr+6 /(x) = -, and g(r) = 5 -3. Then (gojoh)(3) = (22) Let f(x) = r + =, g(r) = 2 2r + 3' and h(x) = v2r. Then (hog of)(4) = (23) Let f(r) = 3(r+ 1)3, g(x) = _ r+ 1 , and h(r) = Vr. Then (ho (g + (h of))) (2) = (24) Let f(x) = r, g(r) = vr + ll, h(x) = 2(r - 1)-1, and j(r) = 4r -1. Then (f o ((hog) + (hoj))) (5) = (25) Let f(x) = r'-br'+ r-7. Find a function g such that (fog)(r) = 2713+9013+78r-2. Answer: g(r) = (26) Let f(x) = cos r and g(r) = r for all r. Write each of the following functions in terms of f and 9. Example. If h(r) = cos'r', then h = go fog. (a) If h(r) = cosr', then h = (b) If h(r) = cosr', then h= (c) If h(r) = cos'r', then h = (d) If h(r) = cos( cos' r), then h = (e) If h(x) = cos'(ri + r'), then h = (27) Let f(r) = r3, g(r) = r - 2, and h(r) = sing for all r. Write each of the following functions in terms of f, 9, and h. Example. If k(r) = sin*(x - 2), then k = fohofog. (a) If k(r) = sin* r, then k = (b) If k(r) = sing , then k = (c) If k(r) = sin(r' - 2), then k = (d) If k(x) = sin(sinr - 2), then k = (e) If k(x) = sin*(sin*(r - 2), then k = (f) If k(r) = sin*(r - 2), then k = (g) If k(x) = sin(r) - 8), then k = (h) If k(r) = sin(r) - 6r' + 12r - 8), then k =10. Let A and B both be the set of all positive integers. Let f : A -> B be defined by f(x) = x. (i) What is the range of this function? Is this function one-to-one? Is it onto? Is it possible to define a one-to-one function from A to B? (i) Can you see that sequences can be thought of as a function f : A -> B where A is the set of all positive integers. E.g. the given function results in the sequence 1,4,9,16, 25.... corresponding to x =1,2.3,4.5. .. 11. Convince yourself that the set of points (x, y) satisfying the following equations all lie on a straight line. In each case, compute the slope and intercept of the line. (i) y = a + hx (ii) y- y. = m(x -x, ) . Show that the line passes through (x,. ), ) (ii) y-y= 1 (x-x, ) . Show that the line passes through (x. ), ) and (x, . ), ). (iv) ay+bx+c=0 For part (iv) describe the line when a =0. Explain why the equation does not represent y as a function of x unless we restrict a # 0. 12.(a) Let f(x) =x . For each of the following, graph g(x) and describe the relationship between g(x) and f (x) (i) g(x) = f (x-a) (ii) g(x)= f(x+a) (ii) g(x)=f (x-a)+b (b) Show that f(x) = ar +hx+ e can be written as fo=afx+b _b -4ac 2a) 40- Show, without using calculus, that (i) the minimum of f (x) occurs at x=-b/2a, (ii) the min. value of f(x) is (4ac-b3) / 4a . (iii) f(x) has one root (i.e. one zero value) if b' =4ac, and two if b' > 4ac Find an expression for the roots of f(x).74. Consider the statements given below and select those ones, which will be correct as per the concept of 'Pigou effect". 1. Wealth effect on the economy. 2. An increase is the real value of the money. 3. Higher demand and higher employment. CODE: (a) 1 and 2 (b) 2 and 3 (c) 1 and 3 (d) 1, 2 and 3Problem 11-4. Stephen has a VNM utility function u(x) = x", where x is his state-contingent wealth. His initial wealth is $160,000. He is considering buying fire insurance, because he faces a 0.05 chance of a small fire that would do $70,000 damage, and a 0.05 chance of a big fire that would do $120,000 damage. He can't suffer both types of fire, i.e. there's a 0.9 chance no fire occurs
