{"id":"cb_question_bank:91364c0e-d7fb-4480-9134-b0fb5cc6cc06","source_id":"cb_question_bank","source_item_id":"91364c0e-d7fb-4480-9134-b0fb5cc6cc06","external_id":"ceae23d4-1965-4201-ac8f-3dbe916cc8e3","ibn":null,"question_id":"23dedddd","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.B.","skill":"Linear functions","difficulty":"M","score_band":4,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In the linear function f , f left parenthesis 0 right parenthesis equals 8 and f left parenthesis 1 right parenthesis equals 12 . Which equation defines f ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">In the linear function <math alttext=\"f\"><mi>f</mi>\n</math>, <math alttext=\"f left parenthesis 0 right parenthesis equals 8\"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mrow><mn>8</mn></mrow></math> and <math alttext=\"f left parenthesis 1 right parenthesis equals 12\"><mi>f</mi><mfenced><mn>1</mn></mfenced><mo>=</mo><mrow><mn>12</mn></mrow></math>. Which equation defines <math alttext=\"f\"><mi>f</mi>\n</math>?</p>","rationale_html":"<p>Choice D is correct. Since <math alttext=\"f\"><mi>f</mi>\n</math> is a linear function, it can be defined by an equation of the form&nbsp;<math alttext=\"f left parenthesis x right parenthesis equals a x plus b\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math>, where <math alttext=\"a\"><mi>a</mi>\n</math> and <math alttext=\"b\"><mi>b</mi>\n</math> are constants. It's given that <math alttext=\"f left parenthesis 0 right parenthesis equals 8\"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>8</mn></math>. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"8\"><mn>8</mn>\n</math> for <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> in the equation <math alttext=\"f left parenthesis x right parenthesis equals a x plus b\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> yields&nbsp;<math alttext=\"8 equals a left parenthesis 0 right parenthesis plus b\"><mn>8</mn><mo>=</mo><mi>a</mi><mfenced><mn>0</mn></mfenced><mo>+</mo><mi>b</mi></math>, or <math alttext=\"8 equals b\"><mn>8</mn><mo>=</mo><mi>b</mi></math>. Substituting <math alttext=\"8\"><mn>8</mn>\n</math> for <math alttext=\"b\"><mi>b</mi>\n</math> in the equation&nbsp;<math alttext=\"f left parenthesis x right parenthesis equals a x plus b\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> yields <math alttext=\"f left parenthesis x right parenthesis equals a x plus 8\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mn>8</mn></math>. It's given that <math alttext=\"f left parenthesis 1 right parenthesis equals 12\"><mi>f</mi><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>12</mn></math>. Substituting <math alttext=\"1\"><mn>1</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"12\"><mn>12</mn>\n</math> for <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> in the equation <math alttext=\"f left parenthesis x right parenthesis equals a x plus 8\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mn>8</mn></math> yields <math alttext=\"12 equals a left parenthesis 1 right parenthesis plus 8\"><mn>12</mn><mo>=</mo><mi>a</mi><mfenced><mn>1</mn></mfenced><mo>+</mo><mn>8</mn></math>, or <math alttext=\"12 equals a plus 8\"><mn>12</mn><mo>=</mo><mi>a</mi><mo>+</mo><mn>8</mn></math>. Subtracting <math alttext=\"8\"><mn>8</mn>\n</math> from both sides of this equation yields <math alttext=\"a equals 4\"><mi>a</mi><mo>=</mo><mn>4</mn></math>. Substituting <math alttext=\"4\"><mn>4</mn>\n</math> for <math alttext=\"a\"><mi>a</mi>\n</math> in the equation&nbsp;<math alttext=\"f left parenthesis x right parenthesis equals a x plus 8\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mn>8</mn></math> yields <math alttext=\"f left parenthesis x right parenthesis equals 4 x plus 8\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>8</mn></math>. Therefore, an equation that defines <math alttext=\"f\"><mi>f</mi>\n</math> is <math alttext=\"f left parenthesis x right parenthesis equals 4 x plus 8\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>8</mn></math>.</p>\n<p>Choice A is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice B 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>","correct_answer":["D"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.812000+00:00","source_updated_at":"2023-08-02T20:25:59.812000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"f left parenthesis x right parenthesis equals 12 x plus 8\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>12</mn></mrow><mi>x</mi><mo>+</mo><mrow><mn>8</mn></mrow></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"f left parenthesis x right parenthesis equals 4 x\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>4</mn></mrow><mi>x</mi></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"f left parenthesis x right parenthesis equals 4 x plus 12\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>4</mn></mrow><mi>x</mi><mo>+</mo><mrow><mn>12</mn></mrow></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"f left parenthesis x right parenthesis equals 4 x plus 8\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mrow><mn>4</mn></mrow><mi>x</mi><mo>+</mo><mrow><mn>8</mn></mrow></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}}