{"id":"cb_question_bank:d7f299a9-d952-49f9-a009-495e01edf5c5","source_id":"cb_question_bank","source_item_id":"d7f299a9-d952-49f9-a009-495e01edf5c5","external_id":null,"ibn":"028940-DC","question_id":"f5e8ccf1","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"M","score_band":4,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The function f is defined above. Which of the following is NOT an x -intercept of the graph of the function in the xy -plane?","stimulus":"f of x equals, open parenthesis, x plus 4, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, times, open parenthesis, 2 x minus 3, close parenthesis","stimulus_html":"<div class=\"stimulus_reference \">\n        <p class=\"standalone_statement style:1 \">\n          <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b51fbc8605fc93cea4ec5bbb2ddd810f5dcd7e3542e4cc7d0ce2a93e028f6873\" alt=\"f of x equals, open parenthesis, x plus 4, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, times, open parenthesis, 2 x minus 3, close parenthesis\"></span></p>\n      </div>\n","stem_html":"<p class=\"stem_paragraph \">The function <span class=\"italic\">f</span> is defined above. Which of the following is NOT an <span class=\"italic \">x</span>-intercept of the graph of the function in the <span class=\"italic \">xy</span>-plane?</p>\n","rationale_html":"<p>Choice B is correct. The graph of the function <span class=\"italic\">f</span> in the <span class=\"italic\">xy</span>-plane has <span class=\"italic\">x</span>-intercepts at the points <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/338db7e3a31d3afebe84d6ca556e60cbe33b3baf49002acc912917c4ff25d31e\" alt=\"with coordinates x comma y\"></span>, where <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0a0da931f88351a15d485b0cf7f4a5d683a3bfcb38b3ad3693a34cf1f53907c6\" alt=\"y equals f of x, which equals 0\"></span>. Substituting 0 for <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/26d9f477bb154c8ef7637e64827776ac5831aa760a31d21a53157367bd210b16\" alt=\"f of x \"></span> in the given equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6a61d3c2f7beb6858baf11cbb1964d30d341e1e9d8524eeda181169ad16309e0\" alt=\"0 equals, open parenthesis, x plus 4, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, times, open parenthesis, 2 x minus 3, close parenthesis\"></span>. By the zero product property, if <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6a61d3c2f7beb6858baf11cbb1964d30d341e1e9d8524eeda181169ad16309e0\" alt=\"0 equals, open parenthesis, x plus 4, close parenthesis, times, open parenthesis, x minus 1, close parenthesis, times, open parenthesis, 2 x minus 3, close parenthesis\"></span>, then <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bedc920ad278563ba967e90e271ee5a22a4065511dc66ee9f8d9ff2498d9b88d\" alt=\"x plus 4, equals 0\"></span>, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5263dc3a6fe5690fe9c6ab41ffc64c8fdeacc4d55faa0e053c52cccb0c306846\" alt=\"x minus 1, equals 0\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b14189b745c4e7b6a04b8c7669ba5e31e946d8fdccd1d52b5c1dc8f817980218\" alt=\"2 x minus 3, equals 0\"></span>. Solving each of these linear equations for <span class=\"italic\">x</span>, it follows that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b2271bf6a6395b1b35156f1d6b258d0ebf35d2f79134117675d045c2f48b0d63\" alt=\"x equals negative 4\"></span>, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6e4319fe2c5755fc5aa3f69e1a43b989d061cbb17c450f24b51a6616f131de4b\" alt=\"x equals 1\"></span>, and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2f9280cb724f9bc73a6cefe3bd4f0836da50ef56ce5749d0fae4a7089dcd5cb2\" alt=\"x equals three halves\"></span>, respectively. This means that the graph of the function <span class=\"italic\">f</span> in the <span class=\"italic\">xy</span>-plane has three <span class=\"italic\">x</span>-intercepts: <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8d5d139a4a8485efe3a3bd9aa2759afb59f2fd9058a3086690b078ac27d784a6\" alt=\"the point with coordinates negative 4 comma 0\"></span>, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/dd517ff65ad53662b01847047d12b83dfb628a81d6983d27b71cf476b5e02ccf\" alt=\"the point with coordinates 1 comma 0\"></span>, and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5b123081e6fa0f39d37faffe5e8d20147976bb46fde8eba86ebe9a5e9f319e85\" alt=\"the point with coordinates three halves comma 0\"></span>. Therefore, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1f51f293120b2380f488160a20556c8362feef4dc1e99998a9ac6d713b4147f3\" alt=\"the point with coordinates negative two thirds comma 0\"></span> isn’t an <span class=\"italic\">x</span>-intercept of the graph of the function <span class=\"italic\">f</span>.<br>\nAlternate approach: Substitution may be used. Since by definition an <span class=\"italic\">x</span>-intercept of any graph is a point in the form <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/dfc7782d84c2d50e207107bed58d6dbb4b91fb95510f30d6bf7a2198e811bcec\" alt=\"k comma 0\"></span> where <span class=\"italic\">k </span>is a constant, and since all points in the options are in this form, it need only be checked whether the points in the options lie on the graph of the function <span class=\"italic\">f</span>. Substituting <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/565a7d173c17435a6cc642c61626cb51e3995546c9ade6015509590e166b0a1d\" alt=\"negative two thirds \"></span> for <span class=\"italic\">x</span> and 0 for <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/26d9f477bb154c8ef7637e64827776ac5831aa760a31d21a53157367bd210b16\" alt=\"f of x\"></span> in the given equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f5a99275a3f2528f97d05710cbd1804f3a974cc0f5ded95d385b662eb6036eb0\" alt=\"0 equals, open parenthesis, negative two thirds plus 4, close parenthesis, times, open parenthesis, negative two thirds minus 1, close parenthesis, times, open parenthesis, 2 times negative two thirds, minus 3, close parenthesis\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b4e1ad877ce0340370ed8b5cb66da1d97af8ece95323c6116a02255340ffdbb3\" alt=\"0 equals, the fraction 650 over 27\"></span>. Therefore, the point <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1f51f293120b2380f488160a20556c8362feef4dc1e99998a9ac6d713b4147f3\" alt=\"with coordinates negative two thirds comma 0 \"></span> doesn’t lie on the graph of the function <span class=\"italic\">f</span> and can’t be an <span class=\"italic\">x</span>-intercept of the graph.</p><p>Choices A, C, and D are incorrect because each of these points is an <span class=\"italic\">x</span>-intercept of the graph of the function <span class=\"italic\">f</span> in the<span class=\"italic\"> xy</span>-plane. By definition, an <span class=\"italic\">x</span>-intercept is a point on the graph of the form <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/dfc7782d84c2d50e207107bed58d6dbb4b91fb95510f30d6bf7a2198e811bcec\" alt=\"k comma 0 \"></span>, where <span class=\"italic\">k</span> is a constant. Substituting <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bc0609d3d5523eaf8e35097c2a9c872ba31206db61297d0a21597123b1c843d7\" alt=\"negative 4 \"></span> for <span class=\"italic\">x</span> and 0 for <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/26d9f477bb154c8ef7637e64827776ac5831aa760a31d21a53157367bd210b16\" alt=\"f of x\"></span> in the given equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b3d2c33ea279e13f7a8d9d73c2b09b474ea6188c5c81c18f79e14c9e84324e22\" alt=\"0 equals, open parenthesis, negative 4 plus 4, close parenthesis, times, open parenthesis, negative 4 minus 1, close parenthesis, times, open parenthesis, 2 times negative 4, minus 3, close parenthesis\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4916ff43d6476b41ff006070114c9bf37a8dd2fa857adf0288c2f5f5c1da1037\" alt=\"0 equals 0\"></span>. Since this is a true statement, the point <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8d5d139a4a8485efe3a3bd9aa2759afb59f2fd9058a3086690b078ac27d784a6\" alt=\"with coordinates negative 4 comma 0 \"></span> lies on the graph of the function <span class=\"italic\">f</span> and is an <span class=\"italic\">x</span>-intercept of the graph. Performing similar substitution using the points <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/dd517ff65ad53662b01847047d12b83dfb628a81d6983d27b71cf476b5e02ccf\" alt=\"with coordinates 1 comma 0\"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5b123081e6fa0f39d37faffe5e8d20147976bb46fde8eba86ebe9a5e9f319e85\" alt=\"three halves comma 0\"></span> also yields the true statement <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4916ff43d6476b41ff006070114c9bf37a8dd2fa857adf0288c2f5f5c1da1037\" alt=\"0 equals 0\"></span>, illustrating that these points also lie on the graph of the function <span class=\"italic\">f </span>and are <span class=\"italic\">x</span>-intercepts of the graph.</p><p></p>\n","correct_answer":["B"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.623000+00:00","source_updated_at":"2023-08-02T20:25:59.623000+00:00","options":[{"label":"A","content_html":"<span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f954a4fdf3298a6426088c0875cc0fd786f5bb7917d9e4a31c3a8bfe0c4bfb3e\" alt=\"the point with coordinates negative 4 comma 0\"></span>\n","ord":0},{"label":"B","content_html":"<span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/cce15d4ebded306eb9ccb29e3eecf16221039d019a5ee4367c12a1b127d5c51d\" alt=\"the point with coordinates negative two thirds comma 0\"></span>\n","ord":1},{"label":"C","content_html":"<span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8374f2c3e4aecc4f70b4e43551f9e18fa9bd5801b88a34a33b7f730e6c6dcbaa\" alt=\"the point with coordinates 1 comma 0\"></span>\n","ord":2},{"label":"D","content_html":"<span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/47fbe24f3bdf273e8a7f3fc4e4e754fd58b85a6c912844c3612d1e8440290f7f\" alt=\"the point with coordinates three halves comma 0\"></span>\n","ord":3}],"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}}