{"id":"cb_question_bank:2edeac8c-7b7f-4af8-aa83-b033b803c8bd","source_id":"cb_question_bank","source_item_id":"2edeac8c-7b7f-4af8-aa83-b033b803c8bd","external_id":null,"ibn":"031669-DC","question_id":"9a144a01","program":"SAT","module":"math","domain_code":"Q","domain":"Problem-Solving and Data Analysis","skill_code":"Q.D.","skill":"Two-variable data: Models and scatterplots","difficulty":"M","score_band":5,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Which of the following is true about the values of 2 to the x power and 2 x plus 2 for x greater than 0 ?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">Which of the following is true about the values of <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4a816b7cd6eaa09e0f495a8a671c6c7fde4b7457ea164b5477a205dfb948bf62\" alt=\"2 to the x power \"></span></span> and <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bac9c39c91424d74bde90a5119178f9979302f602b5a39555d10b6bb7b851e5b\" alt=\"2 x plus 2 \"></span></span> for <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6ca09b244cb4a0414885894bc1ac4e0242305210b23d08ccb20204e38d574daf\" alt=\"x greater than 0\"></span></span>?</p>\n","rationale_html":"<p>Choice C is correct. At <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/37b98ea685bc6992d8e8e7df92ac8c58c1621b437bc601cc6767a21168bc15e0\" alt=\"x equals 0\"></span>, the value of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4a816b7cd6eaa09e0f495a8a671c6c7fde4b7457ea164b5477a205dfb948bf62\" alt=\"2 raised to the x power\"></span> is less than the value of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/74aee07b3665725bc00775858535a53794f09d26b9b227f1b8d7f21b994224a0\" alt=\"2 x plus 2, where 2 to the 0 power is less than, 2 times 0, plus 2\"></span>, which is equivalent to <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/68f8d75846895a71ba1f5b481a08df76d3dbca75ff1e244a3f71e0a8524bc291\" alt=\"1 is less than 2 \"></span>. As the value of <span class=\"italic\">x</span> increases, the value of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4a816b7cd6eaa09e0f495a8a671c6c7fde4b7457ea164b5477a205dfb948bf62\" alt=\"2 raised to the x power\"></span> remains less than the value of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bac9c39c91424d74bde90a5119178f9979302f602b5a39555d10b6bb7b851e5b\" alt=\"2 x plus 2\"></span> until <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ceba1165e53a8baa1cf03f75cde85da34750b8ab2728ef25385d77f80f8276c5\" alt=\"x equals 3\"></span>, which is when the two values are equal: <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f697f12d9c26dd91694c69190923b343432731ebb07fcee63cd92888684135d1\" alt=\"2 cubed equals, 2 times 3, plus 2\"></span>, which is equivalent to <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/40cf8fba9f67b8878470c12ca460bdc86dbde0075cf1ecb427a1e7a861709d94\" alt=\"8 is equal to 8\"></span>. Then, for <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6e63c9fc926c9432310e7e5b36b91facce36c2447501202f3ea592564860fe64\" alt=\"x greater than 3\"></span>, the value of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4a816b7cd6eaa09e0f495a8a671c6c7fde4b7457ea164b5477a205dfb948bf62\" alt=\"2 raised to the x power\"></span> is greater than the value of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bac9c39c91424d74bde90a5119178f9979302f602b5a39555d10b6bb7b851e5b\" alt=\"2 x plus 2\"></span>. So there is a constant, 3, such that when <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/64674ea37e402d5b19192ad8a27489d848493fbd4d2ecd42cbab63c27752c761\" alt=\"0 is less than x, which is less than 3\"></span>, then <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0192a229834481c09ca6a9e12fbfa09beb51c311d3de53acebb0571ebf79a8cd\" alt=\"2 raised to the x power is less than, 2 x plus 2\"></span>, but when <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6e63c9fc926c9432310e7e5b36b91facce36c2447501202f3ea592564860fe64\" alt=\"x is greater than 3\"></span>, then <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5085b9e2d8daf31099fb02e930d0603e988daedc190036d486f782e2aec9371d\" alt=\"2 raised to the x power is greater than, 2 x plus 2\"></span>.</p><p>Choice A is incorrect because <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5085b9e2d8daf31099fb02e930d0603e988daedc190036d486f782e2aec9371d\" alt=\"2 raised to the x power is greater than, 2 x plus 2\"></span> when <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6e63c9fc926c9432310e7e5b36b91facce36c2447501202f3ea592564860fe64\" alt=\"x is greater than 3\"></span>. Choice B is incorrect because <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0192a229834481c09ca6a9e12fbfa09beb51c311d3de53acebb0571ebf79a8cd\" alt=\"2 raised to the x power is less than, 2 x plus 2\"></span> when <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/64674ea37e402d5b19192ad8a27489d848493fbd4d2ecd42cbab63c27752c761\" alt=\"0 is less than x, which is less than 3\"></span>. Choice D is incorrect because <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0192a229834481c09ca6a9e12fbfa09beb51c311d3de53acebb0571ebf79a8cd\" alt=\"2 raised to the x power is less than, 2 x plus 2\"></span> when <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/64674ea37e402d5b19192ad8a27489d848493fbd4d2ecd42cbab63c27752c761\" alt=\"0 is less than x, which is less than 3\"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5085b9e2d8daf31099fb02e930d0603e988daedc190036d486f782e2aec9371d\" alt=\"2 raised to the x power is greater than, 2 x plus 2\"></span> when <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6e63c9fc926c9432310e7e5b36b91facce36c2447501202f3ea592564860fe64\" alt=\"x is greater than 3\"></span>.</p><p>&nbsp;</p><p></p>\n","correct_answer":["C"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.627000+00:00","source_updated_at":"2023-08-02T20:25:59.627000+00:00","options":[{"label":"A","content_html":"<p class=\"choice_paragraph \">For all <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6ca09b244cb4a0414885894bc1ac4e0242305210b23d08ccb20204e38d574daf\" alt=\"x greater than 0\"></span></span>, it is true that <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0192a229834481c09ca6a9e12fbfa09beb51c311d3de53acebb0571ebf79a8cd\" alt=\"2 to the x power, is less than 2 x plus 2\"></span></span>.</p>\n","ord":0},{"label":"B","content_html":"<p class=\"choice_paragraph \">For all <span class=\"math_expression \"><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6ca09b244cb4a0414885894bc1ac4e0242305210b23d08ccb20204e38d574daf\" alt=\"x greater than 0\"></span></span></span>, it is true that <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5085b9e2d8daf31099fb02e930d0603e988daedc190036d486f782e2aec9371d\" alt=\"2 to the x power, is greater than 2 x plus 2\"></span></span>.</p>\n","ord":1},{"label":"C","content_html":"<p class=\"choice_paragraph \">There is a constant <span class=\"italic\">c</span> such that if <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4dd3b9c2c1bc7e0eb736e48757e41040632ccb4a36e60a9df8194a9eaca932a0\" alt=\"0 is less than x, which is less than c\"></span></span>, then <span class=\"math_expression \"><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0192a229834481c09ca6a9e12fbfa09beb51c311d3de53acebb0571ebf79a8cd\" alt=\"2 to the x power, is less than 2 x plus 2\"></span></span></span>, but if <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/364b974902b0b39c05ce83594e5c03acd379b42e9541ce0b364fcaa44881ef8c\" alt=\"x is greater than c\"></span></span>, then <span class=\"math_expression \"><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5085b9e2d8daf31099fb02e930d0603e988daedc190036d486f782e2aec9371d\" alt=\"2 to the x power, is greater than 2 x plus 2\"></span></span></span>.</p>\n","ord":2},{"label":"D","content_html":"<p class=\"choice_paragraph \">There is a constant <span class=\"italic\">c</span> such that if <span class=\"math_expression \"><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4dd3b9c2c1bc7e0eb736e48757e41040632ccb4a36e60a9df8194a9eaca932a0\" alt=\"0 is less than x, which is less than c\"></span></span></span>, then <span class=\"math_expression \"><span class=\"math_expression \"><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5085b9e2d8daf31099fb02e930d0603e988daedc190036d486f782e2aec9371d\" alt=\"2 to the x power, is greater than 2 x plus 2\"></span></span></span></span>, but if <span class=\"math_expression \"><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/364b974902b0b39c05ce83594e5c03acd379b42e9541ce0b364fcaa44881ef8c\" alt=\"x is greater than c\"></span></span></span>, then <span class=\"math_expression \"><span class=\"math_expression \"><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0192a229834481c09ca6a9e12fbfa09beb51c311d3de53acebb0571ebf79a8cd\" alt=\"2 to the x power, is less than 2 x plus 2\"></span></span></span></span>.</p>\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}}