Could the function be linear or exponential or is it neither? Write possible for

Question

Could the function be linear or exponential or is it neither? Write possible formulas for the linear or exponential function.

r 1 3 7 15 31 p(r) 13 19 31 55 103

r 1 3 7 15 31 p(r) 13 19 31 55 103

Explanation / Answer

Linear function can be written as y = ax + b Exponent function can be written as y = aebx a and b are constants Plug in values, r as x and p(r) as y for (r,p(r)) = (1,13) y = ax+b => 13 = a+b ----- "1" for (r,p(r)) = (3,19) => 19 = 3a+b ------ "2" "2" - "1" => 6 = 2a, so a = 3, put 3 back to "1" 13 = 3+b, so b = 10 The linear equation would be y = 3x + 10. Test all other points: (7,31) => 31 = 3*7+10 (15,55) => 55 = 3*15+10 (31,103) => 103 = 3*31+10 since they are all true, y = 3x+10 For exponent function y = aebx(1,13) => 13 =aex ------ "3" (3,19) => 19 = ae3x ------"4" Use "4"/"3" on both side and we get 19/13 = e3x/ex Apply Ln (the natural log on both side) and we will have ln(19/13) = ln(e3x/ex) =ln(e3x)/ln(ex) = 3x/x = 3 But ln(19/13) does not equal to 3, therefore none exponent functionis possible

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