{"id":"cb_question_bank:9331c167-0bf7-4bf5-8f07-b069463b8937","source_id":"cb_question_bank","source_item_id":"9331c167-0bf7-4bf5-8f07-b069463b8937","external_id":"2b6d8220-3a75-42df-a439-6f01d10147a0","ibn":null,"question_id":"84dd43f8","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":6,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"For the function f , f left parenthesis 0 right parenthesis equals 86 , and for each increase in x by 1 , the value of f left parenthesis x right parenthesis decreases by 80 percent sign . What is the value of f left parenthesis 2 right parenthesis ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">For the function <math alttext=\"f\"><mi>f</mi>\n</math>, <math alttext=\"f left parenthesis 0 right parenthesis equals 86\"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>86</mn></math>, and for each increase in <math alttext=\"x\"><mi>x</mi>\n</math> by <math alttext=\"1\"><mn>1</mn>\n</math>, the value of <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> decreases by <math alttext=\"80 percent sign\"><mn>80</mn><mo>%</mo></math>. What is the value of <math alttext=\"f left parenthesis 2 right parenthesis\"><mi>f</mi><mfenced><mn>2</mn></mfenced></math>?</p>","rationale_html":"<p>The correct answer is <math alttext=\"3.44\"><mn>3.44</mn>\n</math>. It’s given that&nbsp;<math alttext=\"f left parenthesis 0 right parenthesis equals 86\"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>86</mn></math> and that for each increase in <math alttext=\"x\"><mi>x</mi>\n</math> by <math alttext=\"1\"><mn>1</mn>\n</math>, the value of&nbsp;<math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> decreases by <math alttext=\"80 percent sign\"><mn>80</mn><mo>%</mo></math>. Because the output of the function decreases by a constant percentage for each <math alttext=\"1\"><mn>1</mn>\n</math>-unit increase in the value of <math alttext=\"x\"><mi>x</mi>\n</math>, this relationship can be represented by an exponential function of the form&nbsp;<math alttext=\"f left parenthesis x right parenthesis equals a left parenthesis b right parenthesis Superscript x\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>a</mi><msup><mfenced><mi>b</mi></mfenced><mi>x</mi></msup></math>, where <math alttext=\"a\"><mi>a</mi>\n</math> represents the initial value of the function and <math alttext=\"b\"><mi>b</mi>\n</math> represents the rate of decay,<br>expressed as a decimal. Because&nbsp;<math alttext=\"f left parenthesis 0 right parenthesis equals 86\"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>86</mn></math>, the value of <math alttext=\"a\"><mi>a</mi>\n</math> must be <math alttext=\"86\"><mn>86</mn>\n</math>. Because the value of&nbsp;<math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> decreases by&nbsp;<math alttext=\"80 percent sign\"><mn>80</mn><mo>%</mo></math> for each <math alttext=\"1\"><mn>1</mn>\n</math>-unit increase in <math alttext=\"x\"><mi>x</mi>\n</math>, the value of <math alttext=\"b\"><mi>b</mi>\n</math> must be <math alttext=\"left parenthesis 1 minus 0 period 80 right parenthesis\"><mo>(</mo><mn>1</mn><mo>−</mo><mn>0</mn><mo>.</mo><mn>80</mn><mo>)</mo></math>, or <math alttext=\"0 period 2\"><mn>0</mn><mo>.</mo><mn>2</mn></math>. Therefore, the function <math alttext=\"f\"><mi>f</mi>\n</math>&nbsp;can be defined by <math alttext=\"f left parenthesis x right parenthesis equals 86 left parenthesis 0 period 2 right parenthesis Superscript x\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>86</mn><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced><mi>x</mi></msup></math>. Substituting <math alttext=\"2\"><mn>2</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> in this function yields <math alttext=\"f left parenthesis 2 right parenthesis equals 86 left parenthesis 0 period 2 right parenthesis squared\"><mi>f</mi><mfenced><mn>2</mn></mfenced><mo>=</mo><mn>86</mn><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></math>, which is equivalent to <math alttext=\"f left parenthesis 2 right parenthesis equals 86 left parenthesis 0 period 04 right parenthesis\"><mi>f</mi><mfenced><mn>2</mn></mfenced><mo>=</mo><mn>86</mn><mfenced><mrow><mn>0</mn><mo>.</mo><mn>04</mn></mrow></mfenced></math>, or <math alttext=\"f left parenthesis 2 right parenthesis equals 3 period 44\"><mi>f</mi><mfenced><mn>2</mn></mfenced><mo>=</mo><mn>3</mn><mo>.</mo><mn>44</mn></math>. Either <math alttext=\"3.44\"><mn>3.44</mn>\n</math> or <math alttext=\"86 divided by 25\"><mn>86</mn><mo>/</mo><mn>25</mn></math>&nbsp;may be entered as the correct answer.</p>\n<p>Alternate approach: It’s given that <math alttext=\"f left parenthesis 0 right parenthesis equals 86\"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>86</mn></math> and that for each increase in <math alttext=\"x\"><mi>x</mi>\n</math> by <math alttext=\"1\"><mn>1</mn>\n</math>, the value of&nbsp;<math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> decreases by <math alttext=\"80 percent sign\"><mn>80</mn><mo>%</mo></math>. Therefore, when&nbsp;<math alttext=\"x equals 1\"><mi>x</mi><mo>=</mo><mn>1</mn></math>, the value of&nbsp;<math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is <math alttext=\"left parenthesis 100 minus 80 right parenthesis percent sign\"><mo>(</mo><mn>100</mn><mo>−</mo><mn>80</mn><mo>)</mo><mo>%</mo></math>, or <math alttext=\"20 percent sign\"><mn>20</mn><mo>%</mo></math>, of <math alttext=\"86\"><mn>86</mn>\n</math>, which can be expressed as <math alttext=\"left parenthesis 0 period 20 right parenthesis left parenthesis 86 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>20</mn></mrow></mfenced><mfenced><mn>86</mn></mfenced></math>. Since&nbsp;<math alttext=\"left parenthesis 0 period 20 right parenthesis left parenthesis 86 right parenthesis equals 17 period 2\"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>20</mn></mrow></mfenced><mfenced><mn>86</mn></mfenced><mo>=</mo><mn>17</mn><mo>.</mo><mn>2</mn></math>, the value of <math alttext=\"f left parenthesis 1 right parenthesis\"><mi>f</mi><mfenced><mn>1</mn></mfenced></math> is <math alttext=\"17.2\"><mn>17.2</mn>\n</math>. Similarly, when&nbsp;<math alttext=\"x equals 2\"><mi>x</mi><mo>=</mo><mn>2</mn></math>, the value of&nbsp;<math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is&nbsp;<math alttext=\"20 percent sign\"><mn>20</mn><mo>%</mo></math> of <math alttext=\"17.2\"><mn>17.2</mn>\n</math>, which can be expressed as <math alttext=\"left parenthesis 0 period 20 right parenthesis left parenthesis 17 period 2 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>20</mn></mrow></mfenced><mfenced><mrow><mn>17</mn><mo>.</mo><mn>2</mn></mrow></mfenced></math>. Since&nbsp;<math alttext=\"left parenthesis 0 period 20 right parenthesis left parenthesis 17 period 2 right parenthesis equals 3 period 44\"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>20</mn></mrow></mfenced><mfenced><mrow><mn>17</mn><mo>.</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>3</mn><mo>.</mo><mn>44</mn></math>, the value of&nbsp;<math alttext=\"f left parenthesis 2 right parenthesis\"><mi>f</mi><mfenced><mn>2</mn></mfenced></math> is <math alttext=\"3.44\"><mn>3.44</mn>\n</math>. Either <math alttext=\"3.44\"><mn>3.44</mn>\n</math> or&nbsp;<math alttext=\"86 divided by 25\"><mn>86</mn><mo>/</mo><mn>25</mn></math> may be entered as the correct answer.</p>","correct_answer":["3.44","86/25"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.825000+00:00","source_updated_at":"2023-08-02T20:25:59.825000+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}}