{"id":"cb_question_bank:c8d03878-6c8c-4a06-a2d8-42654ca8655c","source_id":"cb_question_bank","source_item_id":"c8d03878-6c8c-4a06-a2d8-42654ca8655c","external_id":"9accecc3-e743-4bfa-9e4f-235283d99abd","ibn":null,"question_id":"f1c81b3b","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"M","score_band":5,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The exponential function g is defined by g left parenthesis x right parenthesis equals 19 dot a Superscript x , where a is a positive constant. If g left parenthesis 3 right parenthesis equals 2,375 , what is the value of g left parenthesis 4 right parenthesis ?","stimulus":"","stimulus_html":null,"stem_html":"<p>The exponential function <math alttext=\"g\"><mi>g</mi>\n</math> is defined by <math alttext=\"g left parenthesis x right parenthesis equals 19 dot a Superscript x\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>19</mn></mrow><mo>·</mo><msup><mi>a</mi><mi>x</mi></msup></math>, where <math alttext=\"a\"><mi>a</mi>\n</math> is a positive constant. If <math alttext=\"g left parenthesis 3 right parenthesis equals 2,375\"><mi>g</mi><mfenced><mn>3</mn></mfenced><mo>=</mo><mn>2,375</mn>\n</math>, what is the value of <math alttext=\"g left parenthesis 4 right parenthesis\"><mi>g</mi><mfenced><mrow><mn>4</mn></mrow></mfenced></math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"11,875\"><mn>11,875</mn>\n</math>. It's given that the exponential function <math alttext=\"g\"><mi>g</mi>\n</math> is defined by <math alttext=\"g left parenthesis x right parenthesis equals 19 dot a Superscript x\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>19</mn><mo>·</mo><msup><mi>a</mi><mi>x</mi></msup></math>, where <math alttext=\"a\"><mi>a</mi>\n</math> is a positive constant, and <math alttext=\"g left parenthesis 3 right parenthesis equals 2,375\"><mi>g</mi><mfenced><mn>3</mn></mfenced><mo>=</mo><mn>2,375</mn></math>. It follows that when <math alttext=\"x equals 3\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mn>3</mn>\n</mrow>\n</math>, <math alttext=\"g left parenthesis x right parenthesis equals 2,375\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2,375</mn></math>. Substituting <math alttext=\"3\"><mn>3</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"2,375\"><mn>2,375</mn>\n</math> for&nbsp;<math alttext=\"g left parenthesis x right parenthesis\"><mi>g</mi><mfenced><mi>x</mi></mfenced></math> in the given equation yields <math alttext=\"2,375 equals 19 dot a cubed\"><mn>2,375</mn><mo>=</mo><mn>19</mn><mo>·</mo><msup><mi>a</mi><mn>3</mn></msup></math>. Dividing each side of this equation by <math alttext=\"19\"><mn>19</mn>\n</math> yields <math alttext=\"125 equals a cubed\"><mn>125</mn><mo>=</mo><msup><mi>a</mi><mn>3</mn></msup></math>. Taking the cube root of both sides of this equation gives <math alttext=\"a equals 5\"><mrow>\n\t<mi>a</mi>\n\t<mo>=</mo>\n\t<mn>5</mn>\n</mrow>\n</math>. Substituting <math alttext=\"4\"><mn>4</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"5\"><mn>5</mn>\n</math> for <math alttext=\"a\"><mi>a</mi>\n</math> in the equation&nbsp;<math alttext=\"g left parenthesis x right parenthesis equals 19 dot a Superscript x\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>19</mn><mo>·</mo><msup><mi>a</mi><mi>x</mi></msup></math>&nbsp;yields <math alttext=\"g left parenthesis 4 right parenthesis equals 19 dot 5 Superscript 4\"><mi>g</mi><mfenced><mn>4</mn></mfenced><mo>=</mo><mn>19</mn><mo>·</mo><msup><mn>5</mn><mn>4</mn></msup></math>, or <math alttext=\"g left parenthesis 4 right parenthesis equals 11 comma 875\"><mi>g</mi><mfenced><mn>4</mn></mfenced><mo>=</mo><mn>11</mn><mo>,</mo><mn>875</mn></math>. Therefore, the value of <math alttext=\"g left parenthesis 4 right parenthesis\"><mi>g</mi><mfenced><mn>4</mn></mfenced></math> is <math alttext=\"11,875\"><mn>11,875</mn>\n</math>.&nbsp;</p>","correct_answer":["11875"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.825000+00:00","source_updated_at":"2023-08-02T20:25:59.825000+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}}