Calculus I Homework: Four Ways to Represent a Function Page 1 Questions Example Find f(2 + ), f(x + ), and f(x + ) f(x) were 0 and f(x) = x x 2. Example Find te domain and sketc te grap of te function g(x) = 6 2x. Example Find te domain and sketc te grap of te function f(x) = { 2x + 3 if x < 1 3 x if x 1. Example A taxi company carges two dollars for te first mile (or part of a mile) and 20 cents for eac succeeding tent of a mile (or part). Express te cost C (in dollars) of a ride as a function of te distance traveled (in miles) for 0 < x < 2, and sketc te grap of tis function. Example In a certain country, income tax is assessed as follows. Tere is no tax on income up to $10,000. Any income over $10,000 is taxed at a rate of 10%, up to an income of $20,000. Any income over $20,000 is taxed at 15%. a) Sketc te grap of te tax rate R as a function of te income I. b) How muc tax is assessed on an income of $14,000? On $26,000? c) Sketc te grap of te total assessed tax T as a function of te income I. Solutions Example Find f(2 + ), f(x + ), and f(x + ) f(x) were 0 and f(x) = x x 2. In tis problem we just ave to be careful wit te algebra. Te colour is meant to elp you see ow te functional substitutions work. f(x) = x x 2 f(2 + ) = (2 + ) (2 + ) 2 = (2 + ) (4 + 2 + 4) = 2 2 3 f(x + ) = (x + ) (x + ) 2 = (x + ) (x 2 + 2 + 2x) = x 2 2 2x + x + f(x + ) f(x) = 1 (f(x + ) f(x)) = 1 ( x 2 2 2x + x + ( x x 2)) = 1 ( x 2 2 2x + x + x + x 2) = 1 ( 2 2x + )
Calculus I Homework: Four Ways to Represent a Function Page 2 = ( 2x + 1) = ( 2x + 1), since 0, / = 1. Example Find te domain and sketc te grap of te function g(x) = 6 2x. First, let s get te domain of g(x) = 6 2x. Once we ave te domain, we can construct te grap. We need to use te fact tat te square root function f(y) = y is defined on te real numbers R, only if y 0. For te function g(x), tis means 6 2x 0. 6 2x 0 2x 6 x 3 (dividing by number less tan zero canges te inequality) So te domain of g(x) = 6 2x is x 3. Oter ways of writing te domain are x 3, or x (, 3]. Te range of te function is g(x) [0, ). For te sketc, let s first tink of wat te sketc of te square root function x looks like. Ten I sketced te grap of x. From tese graps we can construct te sketc of g(x). You can ceck tat you grap is correct by making sure it crosses te x and y axis at te proper points. Figure 1: Te steps I used to construct te grap of g(x) = 6 2x. { 2x + 3 if x < 1 Example Find te domain and sketc te grap of te function f(x) = 3 x if x 1. Te function f(x) is piecewise defined. Terefore, I am going to plot all te functions we need individually first, and ten combine tem to get te desired sketc of te grap of f(x). First, let s grap y = 2x+3. Tis is a straigt line, wit te form y = mx+b, were m is te slope and b is te y-intercept. Tis function as a slope of 2, and a y-intercept of 3. To sketc te grap of a straigt line, all we need are two points on te line. Te easiest two points to find are usually te x-intercept and te y-intercept. If x = 0, ten y = 3, so te point (0, 3) is on te line (we knew tis above since we ad worked out te y-intercept!). If y = 0, ten x = 3/2, giving us a second point on te line as ( 3/2, 0). I ve sketced te grap below. We can do te same ting to get a sketc of te grap of y = 3 x. Here we identify te two points (0, 3) and (3, 0) as being on te line.
Calculus I Homework: Four Ways to Represent a Function Page 3 Figure 2: Te grap of y = 2x + 3 and y = 3 x. Now, we combine te above two plots to get te final sketc. It is important to identify were te function canges definition, and label as many points as necessary to make your sketc informative. Note tat I can t easily fill in oles or sow empty oles in my sketces wic were created on a computer. Te best sketc you could present would look someting like te one on te left. Notice tat I ave labeled some points of interest. Figure 3: Te grap of f(x). In tis case we can get te domain and range from te sketc. Tis is different from Problem 1, were we got te domain and range first! Here, te domain is x R, and te range is f(x) 4. Example A taxi company carges two dollars for te first mile (or part of a mile) and 20 cents for eac succeeding tent of a mile (or part). Express te cost C (in dollars) of a ride as a function of te distance traveled (in miles) for 0 < x < 2, and sketc te grap of tis function. In tis problem we are told wat te function is like in words, and need to construct te sketc and analytic form of te function. It is probably easiest to convert te words to a sketc, and ten te sketc to an analytic function. Te analytic function tat relates to tis grap is obviously piecewise defined. You could write a long piecewise definition wic would
Calculus I Homework: Four Ways to Represent a Function Page 4 Figure 4: Te grap of f(x) for te taxi problem. Te computer generated grap is missing te open and filled dots. Your grap sould look more like te one on te left. capture te beaviour of te function in te region 0 < x < 2: f(x) = 2 if x (0, 1] 2.20 if x (1, 1.1] 2.40 if x (1.1, 1.2] 2.60 if x (1.2, 1.3] 2.80 if x (1.3, 1.4] 3.00 if x (1.4, 1.5] 3.20 if x (1.5, 1.6] 3.40 if x (1.6, 1.7] 3.60 if x (1.7, 1.8] 3.80 if x (1.8, 1.9] 4.00 if x (1.9, 2.0]. Tis is very cumbersome. Anoter way or representing te function would be by using te greater integer function, wic we will bump into later. Example In a certain country, income tax is assessed as follows. Tere is no tax on income up to $10,000. Any income over $10,000 is taxed at a rate of 10%, up to an income of $20,000. Any income over $20,000 is taxed at 15%. a) Sketc te grap of te tax rate R as a function of te income I. b) How muc tax is assessed on an income of $14,000? On $26,000? c) Sketc te grap of te total assessed tax T as a function of te income I. Te grap is easily sketced from te information we are given. And from te grap we can obtain te analytic expression for te rate of income tax: 0 if I (0, 10000] Rate of Income Tax = R(I) = 10 if I (10000, 20000] 15 if I (20000, ). To find te tax on an income of $14,000, we do te following:
Calculus I Homework: Four Ways to Represent a Function Page 5 Figure 5: Te grap of tax rate R(I) for te income tax problem. Notice tat tis computer generated grap isn t as good as a and drawn grap because it is missing te open circles and closed dots tat a and drawn grap would ave. Tere sould be circles at (10000, 10) and (20000, 15); tere sould be filled dots at (10000, 0) and (20000, 10). If you look in you text you will see te more correct sketc. 0% tax on te first $10,000 earned = $0 in tax 10% tax on te remaining $4,000 earned = $400 in tax $400 taxes paid To find te tax on an income of $26,000, we do te following: 0% tax on te first $10,000 earned = $0 in tax 10% tax on te next $10,000 earned = $1000 in tax 15% tax on te remaining $6,000 earned = $900 in tax $1900 taxes paid To get a sketc of te assessed tax as a function of income, we need to tink a little bit. For income below $10,000 te tax is assessed at a rate of 0% (no tax is collected). Tis means tat te assessed tax function T (I) will be linear in tis region, and ave a slope of 0. For income from $10,000 to $20,000 te tax is assessed at a rate of 10%. Tis means tat te assessed tax function T (I) will be linear in tis region, and ave a slope of 0.1. For income above $20,000 te tax is assessed at a rate of 15%. Tis means tat te assessed tax function T (I) will be linear in tis region, and ave a slope of 0.15. Tis is combined into te following grap: Getting te analytic function tat tis grap represents involves a small amount of work. I present it ere for tose of you wo want to try and get it. Feel free to stop by if you want some ints! 0 if I (0, 10000] Tax Assessed = T (I) = 0.1I 1000 if I (10000, 20000] 0.15I 2000 if I (20000, ).
Calculus I Homework: Four Ways to Represent a Function Page 6 Figure 6: Te grap of te assessed tax T (I) for te income tax problem.