{"id":"cb_question_bank:c58d8ce5-aff4-4b75-a8c6-70710daea78c","source_id":"cb_question_bank","source_item_id":"c58d8ce5-aff4-4b75-a8c6-70710daea78c","external_id":"e5580e8a-65b2-4cef-9984-856a4c58c355","ibn":null,"question_id":"cd358b89","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Function f is defined by f left parenthesis x right parenthesis equals left parenthesis x plus 6 right parenthesis left parenthesis x plus 5 right parenthesis left parenthesis x plus 1 right parenthesis . Function g is defined by g left parenthesis x right parenthesis equals f left parenthesis x minus 1 right parenthesis . The graph of y equals g left parenthesis x right parenthesis in the xy -plane has x -intercepts at left parenthesis a comma 0 right parenthesis , left parenthesis b comma 0 right parenthesis , and left parenthesis c comma 0 right parenthesis , where a , b , and c are distinct constants. What is the value of a plus b plus c ?","stimulus":"","stimulus_html":null,"stem_html":"

Function f\n is defined by fx=x+6x+5x+1. Function g\n is defined by gx=fx-1. The graph of y=gx in the xy-plane has x-intercepts at a,0, b,0, and c,0, where a\n, b\n, and c\n are distinct constants. What is the value of \n\ta\n\t+\n\tb\n\t+\n\tc\n\n?

","rationale_html":"

Choice B is correct. It's given that gx=fx-1. Since fx=x+6x+5x+1, it follows that fx-1=x-1+6x-1+5x-1+1. Combining like terms yields fx-1=x+5x+4x. Therefore, gx=xx+5x+4. The x-intercepts of a graph in the xy-plane are the points where \n\ty\n\t=\n\t0\n\n. The x-coordinates of the x-intercepts of the graph of y=gx in the xy-plane can be found by solving the equation 0=xx+5x+4. Applying the zero product property to this equation yields three equations: \n\tx\n\t=\n\t0\n\n, \n\t\n\t\tx\n\t\t+\n\t\t5\n\t\n\t=\n\t0\n\n, and \n\t\n\t\tx\n\t\t+\n\t\t4\n\t\n\t=\n\t0\n\n. Solving each of these equations for x\n yields \n\tx\n\t=\n\t0\n\n, \n\tx\n\t=\n\t-5\n\n, and \n\tx\n\t=\n\t-4\n\n, respectively. Therefore, the x-intercepts of the graph of y=gx are 0,0, -5,0, and -4,0. It follows that the values of a\n, b\n, and c\n are 0\n, -5\n, and -4\n. Thus, the value of \n\ta\n\t+\n\tb\n\t+\n\tc\n\n is 0+-5+-4, which is equal to -9\n.

\n

Choice A is incorrect. This is the value of \n\ta\n\t+\n\tb\n\t+\n\tc\n\n if gx=fx+1.

\n

Choice C is incorrect. This is the value of \n\ta\n\t+\n\tb\n\t+\n\tc\n\t-\n\t1\n\n if gx=x-6x-5x-1.

\n

Choice D is incorrect. This is the value of \n\ta\n\t+\n\tb\n\t+\n\tc\n\n if fx=x-6x-5x-1.

","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":"

-15\n

","ord":0},{"label":"B","content_html":"

-9\n

","ord":1},{"label":"C","content_html":"

11\n

","ord":2},{"label":"D","content_html":"

15\n

","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}}