{"id":"cb_question_bank:ea166a5e-c813-4109-85fa-700fdc50edf9","source_id":"cb_question_bank","source_item_id":"ea166a5e-c813-4109-85fa-700fdc50edf9","external_id":null,"ibn":"029650-DC","question_id":"43926bd9","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":6,"answer_type":"spr","answer_source":"rationale","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"x f of x 1 a 2 a, to the fifth power. 3 a, to the ninth power For the exponential function f , the table above shows several values of x and their corresponding values of f of x , where a is a constant greater than 1. If k is a constant and f of k, equals a, to the twenty ninth power , what is the value of k ?","stimulus":"","stimulus_html":null,"stem_html":"<div class=\"stimulus_reference \">\n\t<div class=\"table_wrapper \">\n\t\t<div class=\"tcp-62e5f992-f673-40bd-a548-202c834d3ad3\">\n\t\t\t<table class=\"table_WithBorder tcp-ef7929fc-9906-4a23-8e76-a522e192eb90\"><tbody><tr><td class=\"tcp-02ed6c46-bd1e-4259-860e-081d51202b23\"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8459940867df2424b2ca04535d90ef8626c207ae08e7866b5a032d922cc75eee\" alt=\"x\"></span></td>\n\t\t\t\t\t\t<td class=\"tcp-35ddec34-fbac-495c-ae5b-76f7f7ffab61\"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/83e40a9d411886c8f1db15daf1bfe1768471306e4f8f1a8978edff12e6c8e9e5\" alt=\"f of x\"></span></td>\n\t\t\t\t\t</tr><tr><td class=\"tcp-f28a9eef-359b-45a0-ade4-7c644d87bb6d\"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2f1494c799ca1decbcb94ac7e70aeb91da2c02dfba7cb54dae8e89aee537a8e1\" alt=\"1\"></span></td>\n\t\t\t\t\t\t<td class=\"tcp-775e061f-847b-43ac-97d4-24f063c3ebf2\"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/312ccc3f341afef6fd0d63cb9a21f50d9a0ff66b6f3d8c37bb09f474aae7d490\" alt=\"a\"></span></td>\n\t\t\t\t\t</tr><tr><td class=\"tcp-affd227e-10e3-4509-bf76-780708dbf2d4\"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d870b3a6c89096490c39cc291dea5fd94193a953487d36d45d7f8548ac0b8f4e\" alt=\"2\"></span></td>\n\t\t\t\t\t\t<td class=\"tcp-68d1a00e-9736-470d-bffd-824d15d89e0c\"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/82cefd2b7396261a726049a51e75fca33d0622b0b5c5148bb4f5a7329d52f6d1\" alt=\"a, to the fifth power.\"></span></td>\n\t\t\t\t\t</tr><tr><td class=\"tcp-551cd67d-94b3-42f2-94be-c2adb50ceaf5\"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b35ae18904fb781bd47e663cf02b992965491e4f50b31194e7d69870143879cd\" alt=\"3\"></span></td>\n\t\t\t\t\t\t<td class=\"tcp-4b64f365-e12d-4426-abc5-371a88be077a\"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bbbd25bc16f210be27740344de4c5a41c1d9033daae3c4f214bcbb5039931e55\" alt=\"a, to the ninth power\"></span></td>\n\t\t\t\t\t</tr></tbody></table></div>\n\n\t\t<p><br>\n\t\t\tFor the exponential function <span class=\"italic\">f</span>, the table above shows several values of <span class=\"italic\">x</span> and their corresponding values of <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/26d9f477bb154c8ef7637e64827776ac5831aa760a31d21a53157367bd210b16\" alt=\"f of x\"></span></span>, where <span class=\"italic\">a</span> is a constant greater than 1. If <span class=\"italic\">k</span> is a constant and <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ee2b422f1c5f48188dea708cba0a00c72c28746bfd1107d3ffdf776c04afacf8\" alt=\"f of k, equals a, to the twenty ninth power\"></span></span>, what is the value of <span class=\"italic\">k </span>?</p>\n\t</div>\n</div>\n","rationale_html":"<p>The correct answer is 8. The values of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/26d9f477bb154c8ef7637e64827776ac5831aa760a31d21a53157367bd210b16\" alt=\"f of x \"></span> for the exponential function <span class=\"italic\">f</span> shown in the table increase by a factor of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e5194f03de9c3f577fa36cae1005309c92b1b165939bc5b6eae234bd36719cf6\" alt=\"a, to the fourth power \"></span> for each increase of 1 in <span class=\"italic\">x</span>. This relationship can be represented by the equation <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1966d39d2233094114e62eb08ad051bb78619e18cf273842ea82f7dcccbb2a01\" alt=\"f of x equals, a, raised to the 4 x plus b power\"></span>, where <span class=\"italic\">b</span> is a constant. It’s given that when <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/401169281090f4f1d9aafeeb928524e672d737cdce814b4e9256c5e06a42d325\" alt=\"x equals 2\"></span>, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d4d3b44730d5443ce8df8823efcc2dbb1a6a7a74689e148afb2ee3ea6424f3d0\" alt=\"f of x equals, a, to the fifth power\"></span>.&nbsp; Substituting 2 for <span class=\"italic\">x</span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/82cefd2b7396261a726049a51e75fca33d0622b0b5c5148bb4f5a7329d52f6d1\" alt=\"a, to the fifth power\"></span> for <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/26d9f477bb154c8ef7637e64827776ac5831aa760a31d21a53157367bd210b16\" alt=\"f of x \"></span> into <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1966d39d2233094114e62eb08ad051bb78619e18cf273842ea82f7dcccbb2a01\" alt=\"f of x equals, a, raised to the 4 x plus b power  \"></span> yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5deae8a6e07e5382a436e03dab598e742f97777ae86383b7443bfeca59917d35\" alt=\"a, to the fifth power equals, a, raised to the 4 times 2, plus b power\"></span>. Since <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/9b710ea0a49b6cd51f8eb468495ec5bcd77143bacf2d33ea0aaa0da9118443bc\" alt=\"4 times 2, plus b, equals 5\"></span>, it follows that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f367a7b96248eca6e4cb4c654e02a569bf23acd0d5ead421ff0ca545e6046080\" alt=\"b equals negative 3\"></span>. Thus, an equation that defines the function <span class=\"italic\">f</span> is <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/83b2efe30492d5028af0e33006923d9299184ef8228bf4efd386fd3faf10402c\" alt=\"f of x equals, a, raised to the 4 x minus 3 power\"></span>. It follows that the value of <span class=\"italic\">k</span> such that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cee7e8433da4e9d6546955ac91d6bdec30cd78d1c3ca148c58330e0d55236ad5\" alt=\"f of k equals, a, to the twenty ninth power\"></span> can be found by solving the equation <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c84a9541d37d959b9af60a94d98fc3635a822ec8b6e928f9c41c02eb7494ee7b\" alt=\"4 k minus 3, equals 29\"></span>, which yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2cba1b53ca9de15206e1dbd7cf23bb6ee09a61ec57387b2f53628c9c742c24bd\" alt=\"k equals 8\"></span>.</p>\n","correct_answer":["8"],"has_media":true,"has_mathml":false,"has_table":true,"source_created_at":"2023-08-02T20:25:59.626000+00:00","source_updated_at":"2023-08-02T20:25:59.626000+00:00","options":[],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}