Question

Use the function g(x) = -x+2 with initial values of 4, 2, and 1. What happens after many iterations with all three initial values? How do the results of all three iterations relate to each other?

Answer 1

The given function is , g(x)= - x + 2

`x_(1)=4, g(4)=-4+2=-2,g(-2)=-(-2)+2=4, g(4)=-4+2=-2,g(-2)=-(-2)+2=4`

For values of
`(x_(1))`,
`g(x_(1))` is the sequence, -2,4,-2,4,-2,4........

`x_(2)=2, g(x_(2))=-2+2=0, g(0)=-0+2=2,g(2)=-2+2=0,g(0)-0+2=2\\\\ x_(3)=1,g(x_(3))=-1+2=1, g(1)=-1+2=1,`

For values of
`(x_(2))`,
`g(x_(2))` is the sequence, 0,2,0,2........

For values of
`(x_(3))`,
`g(x_(3))` is the sequence, 1,1,1,1,........

For initial value, of 4,the sequence converges to two values that is -2, 4.

For initial value of 2, the sequence again converges to two values which are 0,2.

For initial value of 1, the sequence converges to one value which is 1.

→→First Iterative value that is (-2)=Second iterative value (0) + 2 ×Third iterative value(1)

→→ First Iterative value that is (4)=Second iterative value (2) + 2 ×Third iterative value(1)