MATH 140 Study Guide - Final Guide: Rational Number, Inverse Function, Radian

Consider the set X, and let F = {f|j: X rightarrow X), the set of functions from X to itself. A notable member of F is, of course, the identity function, id_x: X rightarrow X, defined by the formula: Forall x Element X. id_x(x) = x. In this exercise, we will study the concept of the inverse function. Recall that given two functions f, g Element F we have defined the operation of composition, which is denoted o and takes two functions f, g Element F and produces a new function as follows: f og(x) = f(g(x) for all x Element X. We will give the following definitions: (i) Def 1. A function g Element F if called an left inverse of a function f Element F go f = id_x. (ii) Def 2. A function h Element F if called an right inverse of a function f Element F if f o h = id_x. (iii) Def 3. A function l Element F if called an inverse of a function f Element F if l is simultaneously a left and a right inverse of f. Let LI(f) the predicate "f has a left inverse", RI(f) the predicate "f has a right inverse", and INV(f) the predicate "f has an inverse". Also let INJ(f) be the predicate "f is injective (one-to-one)", SUR(f) "f is surjective (onto)", and "BIJ(f) "f is bijective (one-to-one and onto). Express each of the following statements in predicate logic and write a proof. (i) A function f: X rightarrow X is injective if and only if it has a left inverse. (ii) A function f: X rightarrow X is surjective if and only if it has a right inverse. (iii) A function f: X rightarrow X is bijective if and only if it has an inverse. (iv) If the inverse g of a function f: X rightarrow X exists, it is unique. ("Unique" means the following: if f has another inverse g', then g = g'. Since the inverse of a function is unique, there is a special notation used for it: f ^-1).

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MATH-2 X tation/3/ s bar, Import bockmarks now. SAGE COMLANDS Lab 1: Function Notation Nith SageMath we can easily plot tunctions over specited from-5 to 5 enter the following command intervals of the domain. For example, to graph the function /(r)- + 2r + 7 plot(x"2+2"x+7-5.5 Try this in the SageMath computation cell below we can also graph multiple tunctions at once For example, suppose we want to graph both y-nr) and the transtormaban,-n) .。 on the same axes We do this by frst defining the function r()- +2r +7 and then on the next line we tell SageMath to plot both ) aod fia)+10 on the interval -5.5 f(x)-x-2 , 2.1.7 Try entering these 2 tanes ot code in the Sage Math computation cell below We can specily the color ot each plot as tollows

tion-notation/3 ckmarks bar. Import bookmakes now EVALUATIE We can specity the colot of each plot as follows x)x*2 2 x+7 plot(Tx)Mx+10)-5.S.color Cred black) Try this in the SageMath computation cell above 5 Use the graph of y- fiz) and y-e)+10 above to describe the effect of adding 10 to the function's output 6e(r)-2เว00(ossy is the value of a Chevy Equinox in dollars, t year. after it computation cell below to graph y- ef) and y cit +5) on the interval lo,10, Rememaber, on the first line detine the function e and on the second line use the plot command to graph y n o) and y ci+5) simultaneously When you are finished go to file print, click to yout handout k on more settings and set the scale to 75 and print only page and set Then attach this printout type yout name here s the meaning, in peactical teems od clr +5p Be sure to state the e Use the graphs you printed in ppert (o) to explain the etfect on the grephs hat meanngled units) o·this S of adding S to the angument of the function Pages 123 DOLL DF G

Mnct 60. fx)-3x3x 61, f(x) = x.(x-z? Applications of Differentiation 212 CHAPTER 3 Find all critial hunber 35. 40, g(θ)-48-tane 39. F(x)-x""(x-4): 41. f)-2 cos8 + sino 63. Between 0°C and 30°C, the volume V Gn of I kg of water at a temperature T is 43-44 A formala for the derivartive of a function f'is given. How many critical numbers does f have? V-999.87-0.06426T0.00850437 Find the temperature at which water has its m APTE - 6x + 10 64. An object with weight W is dragged along a h attached to the o by a force acting along a rope rope makes an angle θ with the plane, then the 45-56 Find the absolute maximum and absolute minimurn values the force is μ sin θ + cos θ of f on the given interval. 45. fx)-12+4x -. [0,S] 46. fix)-5+54x-2x. (0,4) 司f(x)-2x,-3x1-12x + 1, [-23] where μ is a positive constant called the coefficie and where 0 θ π/2. Show that F is minimize 65. The water level, measured in feet above mean sea l Lake Lanier in Georgia, USA, during 2012 can be im the function L()-0.01441,, 50. f)- 4). [-2,3] 51.)(02.4) 52. x)-+T -0.4177,2 + 2703t + 1060. where t is measured in months since January 1,201 when the water level was highest during 2012. 66. On May 7, 1992, the space shuttle Endeavour was laun on mission STS 49, the purpose of which was to install perigee kick motor in an Intelsat communications satellit The table gives the velocity data for the shuttle between l and the jettisoning of the solid rocket boosters. (a) Use a graphing calculator or computer to find the cubi S4. ft)- 0.2 pol ymomial that best models the velocity of the shuttle interval r e [o, 125). Then graph this polynom acceleration of the shuttle and usew ss fc)-2 cos+sin 2t, [o, -/2 56. f(t)-' + cot('/2), [π/4.7v/4] 57. If a and b are positive bumbers, find the maximum value 58. Use a graph ximum and minimum values of the to estimate the critical numbers of fx)-II+5xcorect i e decil place 59-62 (a) Use a graph to estimate the absolute (b) Use calculus to find the exact 0 maximurm and 0 185 319 447 values of the function to two decimal places. minimum maxirsum and minimum values 20 32