{"id":"cb_question_bank:c6b2f0f2-8c4b-4354-9abf-710a1a82c445","source_id":"cb_question_bank","source_item_id":"c6b2f0f2-8c4b-4354-9abf-710a1a82c445","external_id":"61063a9c-5dc4-47b2-858c-6c3052f4e804","ibn":null,"question_id":"6efcc0a3","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.B.","skill":"Linear functions","difficulty":"E","score_band":3,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In the linear function h , h left parenthesis 0 right parenthesis equals 41 and h left parenthesis 1 right parenthesis equals 40 . Which equation defines h ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">In the linear function <math alttext=\"h\"><mi>h</mi>\n</math>,&nbsp;<math alttext=\"h left parenthesis 0 right parenthesis equals 41\"><mi>h</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mrow><mn>41</mn></mrow></math> and&nbsp;<math alttext=\"h left parenthesis 1 right parenthesis equals 40\"><mi>h</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>=</mo><mrow><mn>40</mn></mrow></math>. Which equation defines <math alttext=\"h\"><mi>h</mi>\n</math>?</p>","rationale_html":"<p>Choice A is correct. An equation defining a linear function can be written in the form <math alttext=\"h left parenthesis x right parenthesis equals a x plus b\"><mi>h</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=\"h left parenthesis 0 right parenthesis equals 41\"><mi>h</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>41</mn></math>. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"41\"><mn>41</mn>\n</math> for <math alttext=\"h left parenthesis x right parenthesis\"><mi>h</mi><mfenced><mi>x</mi></mfenced></math> in the equation <math alttext=\"h left parenthesis x right parenthesis equals a x plus b\"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> yields <math alttext=\"41 equals a left parenthesis 0 right parenthesis plus b\"><mn>41</mn><mo>=</mo><mi>a</mi><mfenced><mn>0</mn></mfenced><mo>+</mo><mi>b</mi></math>, or <math alttext=\"b equals 41\"><mi>b</mi><mo>=</mo><mn>41</mn></math>. Substituting <math alttext=\"41\"><mn>41</mn>\n</math> for <math alttext=\"b\"><mi>b</mi>\n</math> in the equation <math alttext=\"h left parenthesis x right parenthesis equals a x plus b\"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> yields <math alttext=\"h left parenthesis x right parenthesis equals a x plus 41\"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mn>41</mn></math>. It’s also given that <math alttext=\"h left parenthesis 1 right parenthesis equals 40\"><mi>h</mi><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>40</mn></math>. Substituting <math alttext=\"1\"><mn>1</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"40\"><mn>40</mn>\n</math> for <math alttext=\"h left parenthesis x right parenthesis\"><mi>h</mi><mfenced><mi>x</mi></mfenced></math> in the equation <math alttext=\"h left parenthesis x right parenthesis equals a x plus 41\"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mn>41</mn></math> yields <math alttext=\"40 equals a left parenthesis 1 right parenthesis plus 41\"><mn>40</mn><mo>=</mo><mi>a</mi><mfenced><mn>1</mn></mfenced><mo>+</mo><mn>41</mn></math>, or <math alttext=\"40 equals a plus 41\"><mn>40</mn><mo>=</mo><mi>a</mi><mo>+</mo><mn>41</mn></math>. Subtracting <math alttext=\"41\"><mn>41</mn>\n</math> from the left- and right-hand sides of this equation yields <math alttext=\"negative 1 equals a\"><mo>-</mo><mn>1</mn><mo>=</mo><mi>a</mi></math>. Substituting <math alttext=\"negative 1\"><mo>-</mo><mn>1</mn>\n</math> for <math alttext=\"a\"><mi>a</mi>\n</math> in the equation <math alttext=\"h left parenthesis x right parenthesis equals a x plus 41\"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mn>41</mn></math> yields <math alttext=\"h left parenthesis x right parenthesis equals minus 1 x plus 41\"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mn>1</mn><mi>x</mi><mo>+</mo><mn>41</mn></math>, or <math alttext=\"h left parenthesis x right parenthesis equals negative x plus 41\"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mn>41</mn></math>.</p>\n<p>Choice B is incorrect. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"41\"><mn>41</mn>\n</math> for <math alttext=\"h left parenthesis x right parenthesis\"><mi>h</mi><mfenced><mi>x</mi></mfenced></math> in this equation yields <math alttext=\"41 equals negative 0\"><mn>41</mn><mo>=</mo><mo>-</mo><mn>0</mn></math>, which isn't a true statement.</p>\n<p>Choice C is incorrect. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"41\"><mn>41</mn>\n</math> for <math alttext=\"h left parenthesis x right parenthesis\"><mi>h</mi><mfenced><mi>x</mi></mfenced></math> in this equation yields <math alttext=\"41 equals minus 41 left parenthesis 0 right parenthesis\"><mn>41</mn><mo>=</mo><mo>-</mo><mn>41</mn><mfenced><mn>0</mn></mfenced></math>, or <math alttext=\"41 equals 0\"><mn>41</mn><mo>=</mo><mn>0</mn></math>, which isn't a true statement.</p>\n<p>Choice D is incorrect. Substituting <math alttext=\"41\"><mn>41</mn>\n</math> for <math alttext=\"h left parenthesis x right parenthesis\"><mi>h</mi><mfenced><mi>x</mi></mfenced></math> in this equation yields <math alttext=\"41 equals negative 41\"><mn>41</mn><mo>=</mo><mo>-</mo><mn>41</mn></math>, which isn't a true statement.</p>","correct_answer":["A"],"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=\"h left parenthesis x right parenthesis equals negative x plus 41\"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mrow><mn>41</mn></mrow></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"h left parenthesis x right parenthesis equals negative x\"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>-</mo><mi>x</mi></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"h left parenthesis x right parenthesis equals minus 41 x\"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>-</mo><mrow><mn>41</mn></mrow><mi>x</mi></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"h left parenthesis x right parenthesis equals negative 41\"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>-</mo><mrow><mn>41</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}}