{"id":"cb_question_bank:d17321c1-44d3-437e-9907-0aab26b2a67c","source_id":"cb_question_bank","source_item_id":"d17321c1-44d3-437e-9907-0aab26b2a67c","external_id":"7fc3b869-8f5c-4c0b-a28b-acbe8ea76ddf","ibn":null,"question_id":"0e61101e","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":"f left parenthesis x right parenthesis equals 9 left parenthesis 4 right parenthesis Superscript x The function f is defined by the given equation. If g left parenthesis x right parenthesis equals f left parenthesis x plus 2 right parenthesis , which of the following equations defines the function g ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"f left parenthesis x right parenthesis equals 9 left parenthesis 4 right parenthesis Superscript x\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>9</mn></mrow><msup><mrow><mo>(</mo><mrow><mn>4</mn></mrow><mo>)</mo></mrow><mi>x</mi></msup></math></p>\n<p style=\"text-align: left;\">The function <math alttext=\"f\"><mi>f</mi>\n</math> is defined by the given equation. If <math alttext=\"g left parenthesis x right parenthesis equals f left parenthesis x plus 2 right parenthesis\"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>+</mo><mrow><mn>2</mn></mrow><mo>)</mo></math>, which of the following equations defines the function <math alttext=\"g\"><mi>g</mi>\n</math>?</p>","rationale_html":"<p>Choice B is correct. It’s given that <math alttext=\"f left parenthesis x right parenthesis equals 9 left parenthesis 4 right parenthesis Superscript x\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>9</mn><msup><mfenced><mn>4</mn></mfenced><mi>x</mi></msup></math> and <math alttext=\"g left parenthesis x right parenthesis equals f left parenthesis x plus 2 right parenthesis\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></math>. Substituting&nbsp;<math alttext=\"x plus 2\"><mi>x</mi><mo>+</mo><mn>2</mn></math> for <math alttext=\"x\"><mi>x</mi>\n</math> in&nbsp;<math alttext=\"f left parenthesis x right parenthesis equals 9 left parenthesis 4 right parenthesis Superscript x\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>9</mn><msup><mfenced><mn>4</mn></mfenced><mi>x</mi></msup></math> gives <math alttext=\"f left parenthesis x plus 2 right parenthesis equals 9 left parenthesis 4 right parenthesis Superscript x plus 2\"><mi>f</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>9</mn><msup><mfenced><mn>4</mn></mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></msup></math>. Rewriting this equation using properties of exponents gives <math alttext=\"f left parenthesis x plus 2 right parenthesis equals 9 left parenthesis 4 right parenthesis Superscript x Baseline left parenthesis 4 right parenthesis squared\"><mi>f</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>9</mn><msup><mfenced><mn>4</mn></mfenced><mi>x</mi></msup><msup><mfenced><mn>4</mn></mfenced><mn>2</mn></msup></math>, which is equivalent to <math alttext=\"f left parenthesis x plus 2 right parenthesis equals 9 left parenthesis 4 right parenthesis Superscript x Baseline left parenthesis 16 right parenthesis\"><mi>f</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>9</mn><msup><mfenced><mn>4</mn></mfenced><mi>x</mi></msup><mfenced><mn>16</mn></mfenced></math>. Multiplying <math alttext=\"9\"><mn>9</mn>\n</math> and <math alttext=\"16\"><mn>16</mn>\n</math> in this equation gives <math alttext=\"f left parenthesis x plus 2 right parenthesis equals 144 left parenthesis 4 right parenthesis Superscript x\"><mi>f</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>144</mn><msup><mfenced><mn>4</mn></mfenced><mi>x</mi></msup></math>. Since <math alttext=\"g left parenthesis x right parenthesis equals f left parenthesis x plus 2 right parenthesis\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></math>, <math alttext=\"g left parenthesis x right parenthesis equals 144 left parenthesis 4 right parenthesis Superscript x\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>144</mn><msup><mfenced><mn>4</mn></mfenced><mi>x</mi></msup></math>.</p>\n<p>Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice C is incorrect and may result from conceptual or calculation errors.</p>\n<p>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><math alttext=\"g left parenthesis x right parenthesis equals 18 left parenthesis 4 right parenthesis Superscript x\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>18</mn></mrow><msup><mfenced><mrow><mn>4</mn></mrow></mfenced><mi>x</mi></msup></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"g left parenthesis x right parenthesis equals 144 left parenthesis 4 right parenthesis Superscript x\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>144</mn></mrow><msup><mfenced><mrow><mn>4</mn></mrow></mfenced><mi>x</mi></msup></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"g left parenthesis x right parenthesis equals 18 left parenthesis 8 right parenthesis Superscript x\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>18</mn></mrow><msup><mfenced><mrow><mn>8</mn></mrow></mfenced><mi>x</mi></msup></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"g left parenthesis x right parenthesis equals 81 left parenthesis 16 right parenthesis Superscript x\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>81</mn></mrow><msup><mfenced><mrow><mn>16</mn></mrow></mfenced><mi>x</mi></msup></math></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}}