{"id":"cb_question_bank:516a99c6-7200-4011-94e3-2502a2c2b9ce","source_id":"cb_question_bank","source_item_id":"516a99c6-7200-4011-94e3-2502a2c2b9ce","external_id":"839b7fb0-37a1-46d9-9dce-f3af7e85c97e","ibn":null,"question_id":"b988eeec","program":"SAT","module":"math","domain_code":"H","domain":"Algebra","skill_code":"H.B.","skill":"Linear functions","difficulty":"H","score_band":6,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The functions f and g are defined as f left parenthesis x right parenthesis equals one fourth x minus 9 and g left parenthesis x right parenthesis equals three fourths x plus 21 . If the function h is defined as h left parenthesis x right parenthesis equals f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis , what is the x -coordinate of the x -intercept of the graph of y equals h left parenthesis x right parenthesis in the xy -plane?","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 as <math alttext=\"f left parenthesis x right parenthesis equals one fourth x minus 9\"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac>\n\t<mn>1</mn>\n\t<mn>4</mn>\n</mfrac>\n<mrow>\n\t<mi>x</mi>\n\t<mo>-</mo>\n\t<mn>9</mn>\n</mrow>\n</math> and <math alttext=\"g left parenthesis x right parenthesis equals three fourths x plus 21\"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac>\n\t<mn>3</mn>\n\t<mn>4</mn>\n</mfrac>\n<mrow>\n\t<mi>x</mi>\n\t<mo>+</mo>\n\t<mn>21</mn>\n</mrow>\n</math>. If the function <math alttext=\"h\"><mi>h</mi>\n</math> is defined as <math alttext=\"h left parenthesis x right parenthesis equals f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis\"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, what is the <em>x</em>-coordinate of the <em>x</em>-intercept of the graph of <math alttext=\"y equals h left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>h</mi><mfenced><mi>x</mi></mfenced></math> in the <em>xy</em>-plane?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"negative 12\"><mo>-</mo><mn>12</mn>\n</math>. It's given that the functions <math alttext=\"f\"><mi>f</mi>\n</math> and <math alttext=\"g\"><mi>g</mi>\n</math> are defined as <math alttext=\"f left parenthesis x right parenthesis equals one fourth x minus 9\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo>-</mo><mn>9</mn></math> and <math alttext=\"g left parenthesis x right parenthesis equals three fourths x plus 21\"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>x</mi><mo>+</mo><mn>21</mn></math>. If the function <math alttext=\"h\"><mi>h</mi>\n</math> is defined as <math alttext=\"h left parenthesis x right parenthesis equals f left parenthesis x right parenthesis plus g left parenthesis x right parenthesis\"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>+</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math>, then substituting&nbsp;<math alttext=\"one fourth x minus 9\"><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo>-</mo><mn>9</mn></math> for&nbsp;<math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> and <math alttext=\"three fourths x plus 21\"><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>x</mi><mo>+</mo><mn>21</mn></math> for <math alttext=\"g left parenthesis x right parenthesis\"><mi>g</mi><mfenced><mi>x</mi></mfenced></math>&nbsp;in this function yields <math alttext=\"h left parenthesis x right parenthesis equals one fourth x minus 9 plus three fourths x plus 21\"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo>-</mo><mn>9</mn><mo>+</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>x</mi><mo>+</mo><mn>21</mn></math>. This can be rewritten as <math alttext=\"h left parenthesis x right parenthesis equals four fourths x plus 12\"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>4</mn><mn>4</mn></mfrac><mi>x</mi><mo>+</mo><mn>12</mn></math>, or <math alttext=\"h left parenthesis x right parenthesis equals x plus 12\"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><mo>+</mo><mn>12</mn></math>. The <em>x</em>-intercept of a graph in the <em>xy</em>-plane is the point on the graph where <math alttext=\"y equals 0\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math>. The equation representing the graph of <math alttext=\"y equals h left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>h</mi><mfenced><mi>x</mi></mfenced></math> is <math alttext=\"y equals x plus 12\"><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mn>12</mn></math>. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"y\"><mi>y</mi>\n</math> in this equation yields <math alttext=\"0 equals x plus 12\"><mn>0</mn><mo>=</mo><mi>x</mi><mo>+</mo><mn>12</mn></math>. Subtracting <math alttext=\"12\"><mn>12</mn>\n</math> from both sides of this equation yields <math alttext=\"negative 12 equals x\"><mrow>\n\t<mo>-</mo><mn>12</mn>\n\t<mo>=</mo>\n\t<mi>x</mi>\n</mrow>\n</math>, or <math alttext=\"x equals negative 12\"><mrow>\n\t<mi>x</mi>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>12</mn>\n</mrow>\n</math>. Therefore, the <em>x</em>-coordinate of the <em>x</em>-intercept of the graph of&nbsp;<math alttext=\"y equals h left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>h</mi><mfenced><mi>x</mi></mfenced></math> in the <em>xy</em>-plane is <math alttext=\"negative 12\"><mo>-</mo><mn>12</mn>\n</math>.</p>","correct_answer":["-12"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.811000+00:00","source_updated_at":"2023-08-02T20:25:59.811000+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}}