{"id":1934,"date":"2019-08-25T15:39:15","date_gmt":"2019-08-25T20:39:15","guid":{"rendered":"http:\/\/www.kaptest.com\/blog\/prep\/?p=1934"},"modified":"2021-09-13T16:16:18","modified_gmt":"2021-09-13T16:16:18","slug":"psat-math-functions-defined-by-tables-and-graphs","status":"publish","type":"post","link":"https:\/\/wpapp.kaptest.com\/study\/psat\/psat-math-functions-defined-by-tables-and-graphs\/","title":{"rendered":"PSAT Math: Advanced Functions"},"content":{"rendered":"<p>The ability to interpret the graph of a function will serve you well on Test Day. To interpret graphs of functions, you\u2019ll need to utilize the same skills you use to interpret \u201cregular\u201d equations on the coordinate plane, so this material shouldn\u2019t be completely foreign.<\/p>\n<p>You should know <a title=\"PSAT Math: Functions\" href=\"https:\/\/www.kaptest.com\/blog\/prep\/psat\/psat-math-functions\/\">that a function is merely a dressed-up equation<\/a>, so translating from function to \u201cregular\u201d notation or vice versa is a straightforward process. Consider the following brief example.<br \/>\n<div  class='avia-image-container avia_animated_image avia_animate_when_almost_visible top-to-bottom av-styling-    avia-builder-el-0  el_before_av_icon_box  avia-builder-el-first  avia-align-center '  itemprop=\"image\" itemscope=\"itemscope\" itemtype=\"https:\/\/schema.org\/ImageObject\"  ><div class='avia-image-container-inner'><div class='avia-image-overlay-wrap'><img class='wp-image-0 avia-img-lazy-loading-not-0 avia_image' src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_01-300x300.png\" alt='' title=''   itemprop=\"thumbnailUrl\"  \/><\/div><\/div><\/div><br \/>\nSuppose you\u2019re asked to find the value of <i>x<\/i> for which <i>f<\/i>(<i>x<\/i>) = 6. Because <i>f<\/i>(<i>x<\/i>) represents the output value, or range, translate this as \u201cWhen does the <i>y<\/i> value equal 6?\u201d To answer the question, find 6 on the <i>y<\/i>-axis, then trace over to the function (the line). Read the corresponding <i>x<\/i> value: It\u2019s \u22122, so when <i>f<\/i>(<i>x<\/i>) = 6, <i>x<\/i> must be \u22122.<\/p>\n<p>The PSAT might also present functions in the form of tables. These may or may not have an equation associated with them, but regardless, you\u2019ll need to be adept at extracting the information necessary to answer questions. Most of the time the table will have just two columns, one for the domain and another for the range.<\/p>\n<p><article  class=\"iconbox iconbox_left    avia-builder-el-1  el_after_av_image  el_before_av_heading  \"  itemscope=\"itemscope\" itemtype=\"https:\/\/schema.org\/BlogPosting\" itemprop=\"blogPost\" ><div class=\"iconbox_content\"><header class=\"entry-content-header\"><div class=\"iconbox_icon heading-color\" aria-hidden='true' data-av_icon='\ue864' data-av_iconfont='entypo-fontello'  ><\/div><h3 class='iconbox_content_title  '  itemprop=\"headline\"  >Expert Tip<\/h3><\/header><div class='iconbox_content_container  '  itemprop=\"text\"  ><p>Remember: A value of <i>f<\/i>(<i>x<\/i>) corresponds to a location\u00a0along the <i>y<\/i>-axis. A value of <i>x<\/i> corresponds to a location on the <i>x<\/i>-axis.<\/p>\n<\/div><\/div><footer class=\"entry-footer\"><\/footer><\/article><br \/>\nNow let&#8217;s try a test-like example.<br \/>\n<div  style='padding-bottom:10px; ' class='av-special-heading av-special-heading-h3    avia-builder-el-2  el_after_av_icon_box  el_before_av_promobox  '><h3 class='av-special-heading-tag '  itemprop=\"headline\"  >PSAT Math Practice Question: Functions<\/h3><div class='special-heading-border'><div class='special-heading-inner-border' ><\/div><\/div><\/div><br \/>\n\t<div  style='background:#ffffff;color:#444444;border-color:#444444;' class='av_promobox  avia-button-no   avia-builder-el-3  el_after_av_heading  el_before_av_testimonials '>\t\t<div class='avia-promocontent'><p>\n<a href=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_02_3.png\"><img loading=\"lazy\" class=\"aligncenter size-medium wp-image-1936\" src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_02_3-294x300.png\" alt=\"psat_c09_cb_02_3\" width=\"294\" height=\"300\" \/><\/a><br \/>\n<a href=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/Screen-Shot-2016-08-25-at-2.59.14-PM.png\"><img loading=\"lazy\" class=\"aligncenter size-medium wp-image-1937\" src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/Screen-Shot-2016-08-25-at-2.59.14-PM-300x48.png\" alt=\"PSAT functions math practice \" width=\"300\" height=\"48\" \/><\/a><br \/>\nA. \u22124<br \/>\nB. 6<br \/>\nC. 10<br \/>\nD. 14<\/p>\n<\/div><\/div><\/p>\n<p>Work through the Kaplan Method for Math step-by-step to solve this question. The following table shows Kaplan\u2019s strategic thinking on the left, along with suggested math scratchwork on the right.<\/p>\n<div class=\"s9-scrollable\">\n<table class=\"table table-layout-fixed sat-exclude\">\n<thead>\n<tr>\n<th>Strategic Thinking<\/th>\n<th class=\"no-break\">Math Scratchwork<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr class=\"cell-shade-light\">\n<td><b>Step 1: Read the question, identifying and organizing important information as you go<\/b><br \/>\nYou\u2019re asked to determine the value of f ( 0 ) + g ( 1\/2 ) . In other words, you need to find the <i>y-<\/i>value of function <i>f<\/i> when <i>x<\/i> = 0 and the <i>y-<\/i>value of function <i>g<\/i> when x = 1\/2 , then add those values together.<\/td>\n<td><\/td>\n<\/tr>\n<tr class=\"cell-shade-light\">\n<td class=\"border-bottom-0\"><b>Step 2: Choose the best strategy to answer the question<\/b><br \/>\n<i>How can you find<\/i> f<i>(0)?<\/i><\/p>\n<p>Start with <i>f<\/i>\u200a(<i>x<\/i>). Locate the spot on the graph of\u00a0<i>f<\/i>\u200a(<i>x<\/i>) where <i>x<\/i> = 0. At that point, the\u00a0<i>y<\/i>-value is 4. Therefore,<span class=\"no-break\"> <i>f<\/i>\u200a(0) = 4.<\/span><\/p>\n<p><i>What about <\/i>g<i>(1\/2) ?<\/i><br \/>\nRepeat the previous process for <i>g<\/i>(<i>x<\/i>) when x = 1\/2 ; at this <i>x-<\/i>value, <i>y<\/i> = 10. In function notation, <i>g<\/i>(1\/2) = 10.<br \/>\nThe hard part is over; now just add the values together.<\/td>\n<td class=\"border-top-0\">\n<div class=\"s9-scrollable\">\n<figure class=\"figure\"><img src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/psat_c09_cb_03_3.png\" alt=\"q\" \/><\/figure>\n<\/div>\n<\/td>\n<\/tr>\n<tr class=\"cell-shade-light\">\n<td><b>Step 3: Check that you answered the<\/b> <b><i>right<\/i><\/b> <b>question<\/b><br \/>\nYou found f ( 0 ) + g ( 1\/2 ) , which matches (D). You\u2019re finished.<\/td>\n<td class=\"essay-sample\">\n<i>f<\/i>\u200a(0) + <i>g<\/i>(1\/2) = 4 + 10 = 14<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p><div   data-autoplay='1'  data-interval='5'  data-animation='fade'  data-hoverpause='1'  class='avia-testimonial-wrapper avia-grid-testimonials avia-grid-1-testimonials avia_animate_when_almost_visible   '>\n<section class ='avia-testimonial-row'><div class='avia-testimonial av_one_full flex_column no_margin avia-testimonial-row-1 avia-first-testimonialavia-last-testimonial' ><div class='avia-testimonial_inner'  itemscope=\"itemscope\" itemtype=\"https:\/\/schema.org\/BlogPosting\" itemprop=\"blogPost\" ><div class='avia-testimonial-image'  itemprop=\"image\"  ><\/div><div class='avia-testimonial-content '  ><div class='avia-testimonial-markup-entry-content'  itemprop=\"text\" ><p>Watch your axis scales; just like scatterplot questions, questions involving graphs of functions often contain trap answers for students who misread the axes.<\/p>\n<p>Although\u00a0this question would have been much simpler if the graph had labeled the points or given you an equation to plug values into, it wouldn\u2019t have tested your knowledge of functions. Your ability to figure out what questions about functions are actually asking is key to solving them\u00a0correctly on Test Day.<\/p>\n<\/div><\/div><div class='avia-testimonial-meta'><div class='avia-testimonial-arrow-wrap'><div class='avia-arrow'><\/div><\/div><div class='avia-testimonial-meta-mini'  itemprop=\"author\" itemscope=\"itemscope\" itemtype=\"https:\/\/schema.org\/Person\" ><strong  class='avia-testimonial-name'    itemprop=\"name\" >Laura Aitcheson<\/strong><span  class='avia-testimonial-subtitle '    itemprop=\"jobTitle\" >Kaplan PSAT Expert<\/span><\/div><\/div><\/div><\/div><\/section>\n<\/div><br \/>\n<div  style='padding-bottom:10px; ' class='av-special-heading av-special-heading-h3    avia-builder-el-5  el_after_av_testimonials  el_before_av_image  '><h3 class='av-special-heading-tag '  itemprop=\"headline\"  >Functions on the PSAT: Multiple Functions<\/h3><div class='special-heading-border'><div class='special-heading-inner-border' ><\/div><\/div><\/div><br \/>\nThere are several ways in which the PSAT might ask you to juggle multiple functions simultaneously. Fortunately, the rules governing what to do are easy to understand. To start, we\u2019ll look at how to combine functions. This technique simply involves adding, subtracting, multiplying, and\/or dividing the functions in play. Check out the following table for a synopsis of how to combine functions with the four basic operations (and make them look less intimidating).<\/p>\n<p>On Test Day, you might see <b>piecewise functions<\/b>. A piecewise function is a function that is defined, literally, by multiple pieces. What breaks a function into pieces are different rules that govern different parts of the function\u2019s domain. Here&#8217;s an example:<\/p>\n<div  class='avia-image-container avia_animated_image avia_animate_when_almost_visible top-to-bottom av-styling-    avia-builder-el-6  el_after_av_heading  el_before_av_promobox  avia-align-center '  itemprop=\"image\" itemscope=\"itemscope\" itemtype=\"https:\/\/schema.org\/ImageObject\"  ><div class='avia-image-container-inner'><div class='avia-image-overlay-wrap'><img class='wp-image-0 avia-img-lazy-loading-not-0 avia_image' src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/sat_c10_cb_03-296x300.png\" alt='' title=''   itemprop=\"thumbnailUrl\"  \/><\/div><\/div><\/div>\n<div class=\"s9-scrollable\">fx=1,\u00a0\u00a0\u00a0if\u00a0x\u2264\u20132&#215;2,\u00a0if\u00a0\u20132&lt;x&lt;33,\u00a0\u00a0if\u00a0x=3\u20133x+11,\u00a0\u00a0if\u00a0x&gt;3<\/div>\n<p>In the function shown, the behavior of the graph depends on the domain. Linear, quadratic, and even a single point interval make up this function. Each \u201crule\u201d is written inside the open bracket in &#8220;pieces.&#8221; To the right is the domain interval for which each \u201crule\u201d applies. On the graph, an open dot indicates that a point is not included in the interval; a closed dot indicates one that is. Note that the different inequality signs used in the domain constraints dictate whether a dot is open or closed on the graph. For a single-point interval, an equal sign is used.<\/p>\n<p>To evaluate a piecewise function, first determine to which piece of the domain the input value belongs, and then substitute the value into the corresponding rule. For example, in the function above, <i>f<\/i>(2) = (2)<sup>2<\/sup>\u00a0= 4, because the input value 2 is between \u20132 and 3 (the second piece of the domain).<\/p>\n<p>Similarly, <i>f<\/i>(5) = \u20133(5) + 11 = \u2013\u200a4 because the input value 5 is greater than 3 (the last piece of the domain). You can confirm these values by looking at the graph. At <i>x<\/i>\u00a0= 2, the point on the graph is (2, 4), and at <i>x<\/i>\u00a0= 5, the point on the graph is (5, \u2013\u200a4).<\/p>\n<p>Give this function question a try:<br \/>\n\t<div  style='background:#ffffff;color:#555555;border-color:#444444;' class='av_promobox  avia-button-no   avia-builder-el-7  el_after_av_image  avia-builder-el-last '>\t\t<div class='avia-promocontent'><p>\n<a href=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/Screen-Shot-2016-08-25-at-4.05.24-PM.png\"><img loading=\"lazy\" class=\"  wp-image-1947 alignleft\" src=\"https:\/\/www.kaptest.com\/blog\/prep\/wp-content\/uploads\/sites\/21\/2016\/08\/Screen-Shot-2016-08-25-at-4.05.24-PM-1030x103.png\" alt=\"multiple functions psat practice problem question\" width=\"620\" height=\"62\" \/><\/a><\/p>\n<p>A. \u00a0\u20136<br \/>\nB. \u00a0\u20132<br \/>\nC. \u00a03<br \/>\nD. 9<\/p>\n<\/div><\/div><\/p>\n<p>Appearances can be deceiving. At first glance, this question looks tough, but the following table will clarify anything confusing. Kaplan\u2019s strategic thinking is on the left, along with suggested math scratchwork on the right.<\/p>\n<div class=\"s9-scrollable\">\n<table class=\"table table-layout-fixed sat-exclude\">\n<thead>\n<tr>\n<th>Strategic Thinking<\/th>\n<th>Math Scratchwork<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr class=\"cell-shade-light\">\n<td><b>Step 1: Read the question, identifying and organizing important information as you go<\/b><br \/>\nRead carefully to see what the question is asking. You need to find the product (<i>fg<\/i>) of the results when each function is evaluated at <i>x <\/i>= \u20135.<\/td>\n<td><\/td>\n<\/tr>\n<tr class=\"cell-shade-light\">\n<td><b>Step 2: Choose the best strategy to answer the question<\/b><br \/>\nYou could multiply the two functions together and then evaluate the result, but it&#8217;s quicker to evaluate each function separately and then multiply.<\/td>\n<td class=\"essay-sample border-bottom-0\">f(\u20135)=2(\u20135)+1=\u201310\u00a0+\u00a01=\u20139<\/p>\n<div class=\"s9-scrollable\"><\/div>\n<p>g(\u20135)=(\u20135)+23=\u201333=-1<\/td>\n<\/tr>\n<tr class=\"cell-shade-light\">\n<td><b>Step 3: Check that you answered the<\/b> <b><i>right<\/i><\/b> <b>question<\/b><br \/>\nThe question is asking for the product (<i>fg<\/i>), which is 9, (D).<\/td>\n<td class=\"essay-sample\">\n(<i>f\u200ag<\/i>\u200a)(\u20135) = \u20139\u00a0\u00d7 (\u20131) = 9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>The ability to interpret the graph of a function will serve you well on Test Day. To interpret graphs of functions, you\u2019ll need to utilize the same skills you use to interpret \u201cregular\u201d equations on the coordinate plane, so this material shouldn\u2019t be completely foreign. You should know that a function is merely a dressed-up [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28737,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[240],"tags":[],"_links":{"self":[{"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/posts\/1934"}],"collection":[{"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/comments?post=1934"}],"version-history":[{"count":3,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/posts\/1934\/revisions"}],"predecessor-version":[{"id":38768,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/posts\/1934\/revisions\/38768"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/media\/28737"}],"wp:attachment":[{"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/media?parent=1934"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/categories?post=1934"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpapp.kaptest.com\/study\/wp-json\/wp\/v2\/tags?post=1934"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}