22. (fx - 3+15. find the value of
Answers
Answer:
Cannot be determined, unless assumptions taken.
Any polynomial of order nn requires you to have n+1n+1 data sets/information.
A linear polynomial/line y=ax+by=ax+b requires two points to be defined completely.
A quadratic polynomial/parabola y=ax2+bx+cy=ax2+bx+c requires three points to be defined completely.
So, a 4th order polynomial will require 5 data sets.
If you consider f(x)f(x) in the above question as a 3rd order polynomial (assumption), then the question can be solved.
In general, one may assume f(x)f(x) as a 4th order polynomial and to solve it, assume the coefficient of highest order/leading coefficient as 1. There have been answers to this question in which people have solved it in rote ways. But, I would like to highlight the beauty of the problem by considering a 3 degree polynomial.
Notice one similarity in the question.
The value of the function (independent variable) is equal to the dependent variable for x=1,2,3x=1,2,3.
Now, since f(x)=xf(x)=x is satisfied by x=1,2,3x=1,2,3
and f(x)f(x) is a 3rd order polynomial (assumption).
Define a function-
g(x)=f(x)−xg(x)=f(x)−x
The function g(x)g(x)==00 at x=1,2,3x=1,2,3
which means they are the roots of the function.
Hence,
g(x)=f(x)−x=a(x−1)(x−2)(x−3)g(x)=f(x)−x=a(x−1)(x−2)(x−3)
g(4)=f(4)−4=a∗3!g(4)=f(4)−4=a∗3!
a=2a=2
g(5)=f(5)−5=