{"id":"cb_question_bank:98d80595-4832-4c07-850e-78fafd861a2b","source_id":"cb_question_bank","source_item_id":"98d80595-4832-4c07-850e-78fafd861a2b","external_id":"68f5f6ee-c704-4500-882b-ba9cb2d1a598","ibn":null,"question_id":"01668cd6","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The functions f and g are defined by the given equations, where x greater than or equals 0 . Which of the following equations displays, as a constant or coefficient, the maximum value of the function it defines, where x greater than or equals 0 ? f left parenthesis x right parenthesis equals 33 left parenthesis 0.4 right parenthesis Superscript x plus 3 g left parenthesis x right parenthesis equals 33 left parenthesis 0.16 right parenthesis left parenthesis 0.4 right parenthesis Superscript x minus 2","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">The functions <math alttext=\"f\"><mi>f</mi>\n</math> and <math alttext=\"g\"><mi>g</mi>\n</math> are defined by the given equations, where&nbsp;<math alttext=\"x greater than or equals 0\"><mi>x</mi><mo>≥</mo><mn>0</mn></math>. Which of the following equations displays, as a constant or coefficient, the maximum value of the function it defines, where <math alttext=\"x greater than or equals 0\"><mi>x</mi><mo>≥</mo><mn>0</mn></math>?</p>\n<ol style=\"list-style-type: upper-roman;\">\n<li style=\"text-align: left;\"><math alttext=\"f left parenthesis x right parenthesis equals 33 left parenthesis 0.4 right parenthesis Superscript x plus 3\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>33</mn></mrow><msup><mfenced><mrow><mn>0.4</mn></mrow></mfenced><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msup></math></li>\n<li style=\"text-align: left;\"><math alttext=\"g left parenthesis x right parenthesis equals 33 left parenthesis 0.16 right parenthesis left parenthesis 0.4 right parenthesis Superscript x minus 2\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>33</mn></mrow><mfenced><mrow><mn>0.16</mn></mrow></mfenced><msup><mfenced><mrow><mn>0.4</mn></mrow></mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></msup></math></li>\n</ol>","rationale_html":"<p style=\"text-align: left;\">Choice B is correct. Functions <math alttext=\"f\"><mi>f</mi>\n</math> and <math alttext=\"g\"><mi>g</mi>\n</math> are both exponential functions with a base of <math alttext=\"0.40\"><mn>0.40</mn>\n</math>. Since <math alttext=\"0.40\"><mn>0.40</mn>\n</math> is less than <math alttext=\"1\"><mn>1</mn>\n</math>, functions <math alttext=\"f\"><mi>f</mi>\n</math> and <math alttext=\"g\"><mi>g</mi>\n</math> are both decreasing exponential functions. This means that <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> and&nbsp;<math alttext=\"g left parenthesis x right parenthesis\"><mi>g</mi><mfenced><mi>x</mi></mfenced></math> decrease as <math alttext=\"x\"><mi>x</mi>\n</math> increases. Since&nbsp;<math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> and&nbsp;<math alttext=\"g left parenthesis x right parenthesis\"><mi>g</mi><mfenced><mi>x</mi></mfenced></math> decrease as <math alttext=\"x\"><mi>x</mi>\n</math> increases, the maximum value of each function occurs at the least value of <math alttext=\"x\"><mi>x</mi>\n</math> for which the function is defined. It's given that functions <math alttext=\"f\"><mi>f</mi>\n</math> and <math alttext=\"g\"><mi>g</mi>\n</math> are defined for <math alttext=\"x greater than or equals 0\"><mi>x</mi><mo>≥</mo><mn>0</mn></math>. Therefore, the maximum value of each function occurs at <math alttext=\"x equals 0\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math>. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> in the equation defining <math alttext=\"f\"><mi>f</mi>\n</math> yields <math alttext=\"f left parenthesis 0 right parenthesis equals 33 left parenthesis 0.4 right parenthesis Superscript 0 plus 3\"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>33</mn><msup><mfenced><mn>0.4</mn></mfenced><mrow><mn>0</mn><mo>+</mo><mn>3</mn></mrow></msup></math>, which is equivalent to&nbsp;<math alttext=\"f left parenthesis 0 right parenthesis equals 33 left parenthesis 0.4 right parenthesis cubed\"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>33</mn><msup><mfenced><mn>0.4</mn></mfenced><mn>3</mn></msup></math>, or&nbsp;<math alttext=\"f left parenthesis 0 right parenthesis equals 2.112\"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>2.112</mn></math>. Therefore, the maximum value of <math alttext=\"f\"><mi>f</mi>\n</math> is <math alttext=\"2.112\"><mn>2.112</mn>\n</math>. Since the equation <math alttext=\"f left parenthesis x right parenthesis equals 33 left parenthesis 0.4 right parenthesis Superscript x plus 3\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>33</mn><msup><mfenced><mn>0.4</mn></mfenced><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msup></math> doesn't display the value <math alttext=\"2.112\"><mn>2.112</mn>\n</math>, the equation defining <math alttext=\"f\"><mi>f</mi>\n</math> doesn't display the maximum value of <math alttext=\"f\"><mi>f</mi>\n</math>. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> in the equation defining <math alttext=\"g\"><mi>g</mi>\n</math> yields <math alttext=\"g left parenthesis 0 right parenthesis equals 33 left parenthesis 0.16 right parenthesis left parenthesis 0.4 right parenthesis Superscript 0 minus 2\"><mi>g</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>33</mn><mfenced><mn>0.16</mn></mfenced><msup><mfenced><mn>0.4</mn></mfenced><mrow><mn>0</mn><mo>-</mo><mn>2</mn></mrow></msup></math>, which can be rewritten as <math alttext=\"g left parenthesis 0 right parenthesis equals 33 left parenthesis 0.16 right parenthesis left parenthesis StartFraction 1 Over 0.4 squared EndFraction right parenthesis\"><mi>g</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>33</mn><mfenced><mn>0.16</mn></mfenced><mfenced><mfrac><mn>1</mn><msup><mn>0.4</mn><mn>2</mn></msup></mfrac></mfenced></math>, or&nbsp;<math alttext=\"g left parenthesis 0 right parenthesis equals 33 left parenthesis 0.16 right parenthesis left parenthesis StartFraction 1 Over 0.16 EndFraction right parenthesis\"><mi>g</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>33</mn><mfenced><mn>0.16</mn></mfenced><mfenced><mfrac><mn>1</mn><mn>0.16</mn></mfrac></mfenced></math>, which is equivalent to&nbsp;<math alttext=\"g left parenthesis 0 right parenthesis equals 33\"><mi>g</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>33</mn></math>. Therefore, the maximum value of <math alttext=\"g\"><mi>g</mi>\n</math> is <math alttext=\"33\"><mn>33</mn>\n</math>. Since the equation&nbsp;<math alttext=\"g left parenthesis x right parenthesis equals 33 left parenthesis 0.16 right parenthesis left parenthesis 0.4 right parenthesis Superscript x minus 2\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>33</mn><mfenced><mn>0.16</mn></mfenced><msup><mfenced><mn>0.4</mn></mfenced><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow></msup></math> displays the value <math alttext=\"33\"><mn>33</mn>\n</math>, the equation defining <math alttext=\"g\"><mi>g</mi>\n</math> displays the maximum value of <math alttext=\"g\"><mi>g</mi>\n</math>. Thus, only equation II displays, as a constant or coefficient, the maximum value of the function it defines.</p>\n<p style=\"text-align: left;\">Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice C is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["B"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.828000+00:00","source_updated_at":"2023-08-02T20:25:59.828000+00:00","options":[{"label":"A","content_html":"<p style=\"text-align: left;\">I only</p>","ord":0},{"label":"B","content_html":"<p style=\"text-align: left;\">II only</p>","ord":1},{"label":"C","content_html":"<p style=\"text-align: left;\">I and II</p>","ord":2},{"label":"D","content_html":"<p style=\"text-align: left;\">Neither I nor II</p>","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}}