{"id":137,"date":"2023-06-21T13:22:37","date_gmt":"2023-06-21T13:22:37","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/putting-it-together-relations-and-functions\/"},"modified":"2023-07-09T02:56:19","modified_gmt":"2023-07-09T02:56:19","slug":"putting-it-together-relations-and-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/putting-it-together-relations-and-functions\/","title":{"raw":"Putting It Together: Function Basics","rendered":"Putting It Together: Function Basics"},"content":{"raw":"\n\nAt the beginning of the module, you were considering Galileo\u2019s famous experiment in which he dropped, or thought about dropping, two balls of different masses from the top of the Tower of Pisa. And you looked at data describing a falling object. Now that you learned about functions, let\u2019s take another look at the data.\n<div>\n<table style=\"width: 50%;\">\n<tbody>\n<tr>\n<td>Time (s)<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>Velocity (m\/s)<\/td>\n<td>0<\/td>\n<td>9.8<\/td>\n<td>19.6<\/td>\n<td>29.4<\/td>\n<td>39.2<\/td>\n<td>49.0<\/td>\n<td>58.8<\/td>\n<td>68.6<\/td>\n<td>78.4<\/td>\n<\/tr>\n<tr>\n<td>Distance (m)<\/td>\n<td>0<\/td>\n<td>4.9<\/td>\n<td>19.6<\/td>\n<td>44.1<\/td>\n<td>78.4<\/td>\n<td>122.5<\/td>\n<td>176.4<\/td>\n<td>240.1<\/td>\n<td>313.6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p style=\"text-align: center;\"><em>Chart shows the correlation between time (seconds 1-8), velocity and distance.<\/em><\/p>\n&nbsp;\n\nConsider the set of ordered pairs relating time to velocity.\n<p style=\"text-align: center;\">{(0, 0), (1, 9.8), (2, 19.6), (3, 29.4), (4, 39.2), (5, 49.0), (6, 58.8), (7, 68.6), and (8, 78.4)}<\/p>\n&nbsp;\n\nIs the relation a function? Indeed it is. Each input value corresponds to only one output value.\n\n<img class=\"wp-image-3698 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/03\/17161132\/Screen-Shot-2017-03-17-at-9.11.07-AM-300x290.png\" alt=\"Graph shows inputs from 0 to 8, and how outputs correlate with each input.\" width=\"387\" height=\"374\">\n\n&nbsp;\n\nNow consider the set of ordered pairs relating time to distance.\n<p style=\"text-align: center;\">{(0, 0), (1, 4.9), (2, 19.6), (3, 44.1), (4, 78.4), (5, 122.5), (6, 176.4), (7, 240.1), and (8, 313.6)}<\/p>\n&nbsp;\n\nAs before, this relation is also a function.\n<img class=\" wp-image-3700 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/03\/17161528\/Screen-Shot-2017-03-17-at-9.13.07-AM-295x300.png\" alt=\"Graph shows inputs from 0 to 8, and how outputs correlate with each input.\" width=\"385\" height=\"392\">\n\n&nbsp;\n\nNow that you know that both velocity and distance can be described by functions, you can evaluate them from the table.\n<ul>\n \t<li style=\"font-weight: 400;\">Suppose velocity as a function of time is represented as [latex]V(t)[\/latex]. &nbsp;Evaluate and explain [latex]V(4)[\/latex]. To evaluate, find the value of the function at 4 seconds. &nbsp;At 4 seconds, velocity is 39.2 m\/s. &nbsp;Therefore, [latex]V(4)=39.2[\/latex].<\/li>\n<\/ul>\n<ul>\n \t<li style=\"font-weight: 400;\">Similarly, solve for [latex]t[\/latex] when [latex]V(t)=49.0[\/latex]. &nbsp;Find 49.0 m\/s on the table and read the related time, 5 s. That means that the velocity is 49.0 m\/s after 5 seconds so [latex]V(5)=49.0[\/latex].<\/li>\n<\/ul>\n<ul>\n \t<li style=\"font-weight: 400;\">Suppose now that the distance is represented as [latex]D(t)[\/latex]. &nbsp;Evaluate and explain [latex]D(6)[\/latex]. To evaluate, find the value of the function at 6 seconds. It is 176.4 m, which means that the falling object travels 176.4 m in 6 seconds.<\/li>\n<\/ul>\n<ul>\n \t<li style=\"font-weight: 400;\">We can also solve [latex]D(t)=240.1[\/latex]. &nbsp;Find 240.1 m in the table and read the related time, which is 7 seconds. &nbsp;So [latex]D(7)=240.1[\/latex].<\/li>\n<\/ul>\n&nbsp;\n\nYou can also use the table to try to determine the function formulas. Begin by reading across the velocity data to look for a pattern. Each value is 9.8 times the number of seconds. &nbsp;So the velocity as a function of time can be represented by the formula [latex]V(t)=at[\/latex], where a is acceleration (in this case due to gravity) and [latex]t[\/latex] is time.\n\nFinding the function formula for distance is a little trickier. The distance as a function of time can be represented by the formula , where [latex]a[\/latex] is acceleration and [latex]t[\/latex] is time.\n\nNow that you know the function formulas, you can evaluate them for any value of time. &nbsp;Suppose it takes an object dropped from a higher floor of the Burj Khalifa 10 seconds to fall to the ground. How fast would the object be traveling when it hit, and how far would it have fallen in that time?\n\nWe can evaluate the functions for velocity and distance at 10 seconds to answer these questions.\n\nSubstitute 10 seconds for [latex]t[\/latex] in each function. Remember that acceleration due to gravity is 9.8 m\/s<sup>2<\/sup>.\n\n&nbsp;\n<div>\n<table style=\"width: 40%;\">\n<tbody>\n<tr>\n<td>[latex]V(t)=at[\/latex]<\/td>\n<td>[latex]D(10)={\\Large\\frac{1}{2}}at^2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]V(10)=a(10)[\/latex]<\/td>\n<td>[latex]D(10)={\\Large\\frac{1}{2}}a(10)^2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]V(10)=(9.8)(10)[\/latex]<\/td>\n<td>[latex]D(10)={\\Large\\frac{1}{2}}(9.8)(10)^2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]V(10)=98[\/latex]<\/td>\n<td>[latex]D(10)=490[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n&nbsp;\n\nAdd these values to the table to the table and what do you see? Velocity and distance really start to increase the longer the object falls. Let\u2019s evaluate the functions at 13 seconds and add those results to the table as well.\n\n&nbsp;\n<div>\n<table style=\"width: 50%;\">\n<tbody>\n<tr>\n<td>Time (s)<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>8<\/td>\n<td><span style=\"color: #ff0000;\">10<\/span><\/td>\n<td><span style=\"color: #ff0000;\">13<\/span><\/td>\n<\/tr>\n<tr>\n<td>Velocity (m\/s)<\/td>\n<td>0<\/td>\n<td>9.8<\/td>\n<td>19.6<\/td>\n<td>29.4<\/td>\n<td>39.2<\/td>\n<td>49.0<\/td>\n<td>58.8<\/td>\n<td>68.6<\/td>\n<td>78.4<\/td>\n<td><span style=\"color: #ff0000;\">98<\/span><\/td>\n<td><span style=\"color: #ff0000;\">127.4<\/span><\/td>\n<\/tr>\n<tr>\n<td>Distance (m)<\/td>\n<td>0<\/td>\n<td>4.9<\/td>\n<td>19.6<\/td>\n<td>44.1<\/td>\n<td>78.4<\/td>\n<td>122.5<\/td>\n<td>176.4<\/td>\n<td>240.1<\/td>\n<td>313.6<\/td>\n<td><span style=\"color: #ff0000;\">490<\/span><\/td>\n<td><span style=\"color: #ff0000;\">828.1<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p style=\"text-align: center;\"><em>Chart shows the correlation between time (seconds 1-8, 10 and 13), velocity and distance.<\/em><\/p>\n&nbsp;\n\nThat means that it would take about 13 seconds for an object to fall the tremendous height of the tallest building in the world, and it would be traveling about 127 m\/s. That is almost 300 miles per hour! You certainly don\u2019t want to be under that when it falls. Thankfully, due to your knowledge of functions, you would know to stay far away.\n\n&nbsp;\n\n","rendered":"<p>At the beginning of the module, you were considering Galileo\u2019s famous experiment in which he dropped, or thought about dropping, two balls of different masses from the top of the Tower of Pisa. And you looked at data describing a falling object. Now that you learned about functions, let\u2019s take another look at the data.<\/p>\n<div>\n<table style=\"width: 50%;\">\n<tbody>\n<tr>\n<td>Time (s)<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>Velocity (m\/s)<\/td>\n<td>0<\/td>\n<td>9.8<\/td>\n<td>19.6<\/td>\n<td>29.4<\/td>\n<td>39.2<\/td>\n<td>49.0<\/td>\n<td>58.8<\/td>\n<td>68.6<\/td>\n<td>78.4<\/td>\n<\/tr>\n<tr>\n<td>Distance (m)<\/td>\n<td>0<\/td>\n<td>4.9<\/td>\n<td>19.6<\/td>\n<td>44.1<\/td>\n<td>78.4<\/td>\n<td>122.5<\/td>\n<td>176.4<\/td>\n<td>240.1<\/td>\n<td>313.6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p style=\"text-align: center;\"><em>Chart shows the correlation between time (seconds 1-8), velocity and distance.<\/em><\/p>\n<p>&nbsp;<\/p>\n<p>Consider the set of ordered pairs relating time to velocity.<\/p>\n<p style=\"text-align: center;\">{(0, 0), (1, 9.8), (2, 19.6), (3, 29.4), (4, 39.2), (5, 49.0), (6, 58.8), (7, 68.6), and (8, 78.4)}<\/p>\n<p>&nbsp;<\/p>\n<p>Is the relation a function? Indeed it is. Each input value corresponds to only one output value.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3698 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/03\/17161132\/Screen-Shot-2017-03-17-at-9.11.07-AM-300x290.png\" alt=\"Graph shows inputs from 0 to 8, and how outputs correlate with each input.\" width=\"387\" height=\"374\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Now consider the set of ordered pairs relating time to distance.<\/p>\n<p style=\"text-align: center;\">{(0, 0), (1, 4.9), (2, 19.6), (3, 44.1), (4, 78.4), (5, 122.5), (6, 176.4), (7, 240.1), and (8, 313.6)}<\/p>\n<p>&nbsp;<\/p>\n<p>As before, this relation is also a function.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3700 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/03\/17161528\/Screen-Shot-2017-03-17-at-9.13.07-AM-295x300.png\" alt=\"Graph shows inputs from 0 to 8, and how outputs correlate with each input.\" width=\"385\" height=\"392\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Now that you know that both velocity and distance can be described by functions, you can evaluate them from the table.<\/p>\n<ul>\n<li style=\"font-weight: 400;\">Suppose velocity as a function of time is represented as [latex]V(t)[\/latex]. &nbsp;Evaluate and explain [latex]V(4)[\/latex]. To evaluate, find the value of the function at 4 seconds. &nbsp;At 4 seconds, velocity is 39.2 m\/s. &nbsp;Therefore, [latex]V(4)=39.2[\/latex].<\/li>\n<\/ul>\n<ul>\n<li style=\"font-weight: 400;\">Similarly, solve for [latex]t[\/latex] when [latex]V(t)=49.0[\/latex]. &nbsp;Find 49.0 m\/s on the table and read the related time, 5 s. That means that the velocity is 49.0 m\/s after 5 seconds so [latex]V(5)=49.0[\/latex].<\/li>\n<\/ul>\n<ul>\n<li style=\"font-weight: 400;\">Suppose now that the distance is represented as [latex]D(t)[\/latex]. &nbsp;Evaluate and explain [latex]D(6)[\/latex]. To evaluate, find the value of the function at 6 seconds. It is 176.4 m, which means that the falling object travels 176.4 m in 6 seconds.<\/li>\n<\/ul>\n<ul>\n<li style=\"font-weight: 400;\">We can also solve [latex]D(t)=240.1[\/latex]. &nbsp;Find 240.1 m in the table and read the related time, which is 7 seconds. &nbsp;So [latex]D(7)=240.1[\/latex].<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p>You can also use the table to try to determine the function formulas. Begin by reading across the velocity data to look for a pattern. Each value is 9.8 times the number of seconds. &nbsp;So the velocity as a function of time can be represented by the formula [latex]V(t)=at[\/latex], where a is acceleration (in this case due to gravity) and [latex]t[\/latex] is time.<\/p>\n<p>Finding the function formula for distance is a little trickier. The distance as a function of time can be represented by the formula , where [latex]a[\/latex] is acceleration and [latex]t[\/latex] is time.<\/p>\n<p>Now that you know the function formulas, you can evaluate them for any value of time. &nbsp;Suppose it takes an object dropped from a higher floor of the Burj Khalifa 10 seconds to fall to the ground. How fast would the object be traveling when it hit, and how far would it have fallen in that time?<\/p>\n<p>We can evaluate the functions for velocity and distance at 10 seconds to answer these questions.<\/p>\n<p>Substitute 10 seconds for [latex]t[\/latex] in each function. Remember that acceleration due to gravity is 9.8 m\/s<sup>2<\/sup>.<\/p>\n<p>&nbsp;<\/p>\n<div>\n<table style=\"width: 40%;\">\n<tbody>\n<tr>\n<td>[latex]V(t)=at[\/latex]<\/td>\n<td>[latex]D(10)={\\Large\\frac{1}{2}}at^2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]V(10)=a(10)[\/latex]<\/td>\n<td>[latex]D(10)={\\Large\\frac{1}{2}}a(10)^2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]V(10)=(9.8)(10)[\/latex]<\/td>\n<td>[latex]D(10)={\\Large\\frac{1}{2}}(9.8)(10)^2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]V(10)=98[\/latex]<\/td>\n<td>[latex]D(10)=490[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Add these values to the table to the table and what do you see? Velocity and distance really start to increase the longer the object falls. Let\u2019s evaluate the functions at 13 seconds and add those results to the table as well.<\/p>\n<p>&nbsp;<\/p>\n<div>\n<table style=\"width: 50%;\">\n<tbody>\n<tr>\n<td>Time (s)<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<td>5<\/td>\n<td>6<\/td>\n<td>7<\/td>\n<td>8<\/td>\n<td><span style=\"color: #ff0000;\">10<\/span><\/td>\n<td><span style=\"color: #ff0000;\">13<\/span><\/td>\n<\/tr>\n<tr>\n<td>Velocity (m\/s)<\/td>\n<td>0<\/td>\n<td>9.8<\/td>\n<td>19.6<\/td>\n<td>29.4<\/td>\n<td>39.2<\/td>\n<td>49.0<\/td>\n<td>58.8<\/td>\n<td>68.6<\/td>\n<td>78.4<\/td>\n<td><span style=\"color: #ff0000;\">98<\/span><\/td>\n<td><span style=\"color: #ff0000;\">127.4<\/span><\/td>\n<\/tr>\n<tr>\n<td>Distance (m)<\/td>\n<td>0<\/td>\n<td>4.9<\/td>\n<td>19.6<\/td>\n<td>44.1<\/td>\n<td>78.4<\/td>\n<td>122.5<\/td>\n<td>176.4<\/td>\n<td>240.1<\/td>\n<td>313.6<\/td>\n<td><span style=\"color: #ff0000;\">490<\/span><\/td>\n<td><span style=\"color: #ff0000;\">828.1<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p style=\"text-align: center;\"><em>Chart shows the correlation between time (seconds 1-8, 10 and 13), velocity and distance.<\/em><\/p>\n<p>&nbsp;<\/p>\n<p>That means that it would take about 13 seconds for an object to fall the tremendous height of the tallest building in the world, and it would be traveling about 127 m\/s. That is almost 300 miles per hour! You certainly don\u2019t want to be under that when it falls. Thankfully, due to your knowledge of functions, you would know to stay far away.<\/p>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-137\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Putting It Together: Function Basics. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Input vs Output (1). <strong>Authored by<\/strong>: Christine Caputo for Lumen. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Input vs Output (2). <strong>Authored by<\/strong>: Christine Caputo for Lumen. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":36,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Putting It Together: Function Basics\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Input vs Output (1)\",\"author\":\"Christine Caputo for Lumen\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Input vs Output (2)\",\"author\":\"Christine Caputo for Lumen\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"43db88f4-d545-4566-a0c6-11f5c1c1cb4a","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-137","chapter","type-chapter","status-publish","hentry"],"part":115,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/137","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/137\/revisions"}],"predecessor-version":[{"id":632,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/137\/revisions\/632"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/115"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/137\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=137"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=137"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=137"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}