Final answer:
After substituting the given x-values into the provided cubic equation options, none of the options match all the given f(x) values. Therefore, the correct cubic equation is not listed among the provided choices.
Step-by-step explanation:
The student is asked to find the cubic equation given four values of the function f(x). To find the correct equation, we substitute each given x-value into the possible cubic equations and check which one matches the corresponding f(x) values. Let's check each option one by one:
For f(-1)=3: a) f(-1) = (-1)^3 - 2(-1)^2 + 3(-1) + 1 = -1 - 2 - 3 + 1 = -5 (does not match) b) f(-1) = (-1)^3 + 2(-1)^2 + 3(-1) + 1 = -1 + 2 - 3 + 1 = -1 (does not match) c) f(-1) = (-1)^3 - 2(-1)^2 + 3(-1) - 1 = -1 - 2 - 3 - 1 = -7 (does not match) d) f(-1) = (-1)^3 + 2(-1)^2 + 3(-1) - 1 = -1 + 2 - 3 - 1 = -3 (does not match)For f(1)=1: a) f(1) = (1)^3 - 2(1)^2 + 3(1) + 1 = 1 - 2 + 3 + 1 = 3 (does not match) b) f(1) = (1)^3 + 2(1)^2 + 3(1) + 1 = 1 + 2 + 3 + 1 = 7 (does not match) c) f(1) = (1)^3 - 2(1)^2 + 3(1) - 1 = 1 - 2 + 3 - 1 = 1 (matches) d) f(1) = (1)^3 + 2(1)^2 + 3(1) - 1 = 1 + 2 + 3 - 1 = 5 (does not match)For f(2)=6 and f(3)=7, we only need to check option c) since it's the only one that matched so far, and we find: c) f(2) = (2)^3 - 2(2)^2 + 3(2) - 1 = 8 - 8 + 6 - 1 = 5 (does not match) c) f(3) = (3)^3 - 2(3)^2 + 3(3) - 1 = 27 - 18 + 9 - 1 = 17 (does not match)
None of the provided options match all the given values, so the correct cubic equation cannot be determined from the choices given.