Given an initial value 𝑥 sub zero, we can iterate a function using a recursive formula. 𝑥 sub 𝑖 equals 𝑓 of 𝑥 sub 𝑖 minus one. Find 𝑥 sub three, given the function 𝑓 of 𝑥 equals three 𝑥 minus two with an initial value of 𝑥 sub zero equals two.
So our initial value is 𝑥 sub zero equals two. So if we would like to find 𝑥 sub one, then we need to plug in one for 𝑖. And one minus one is zero. So we need to plug in 𝑓 of 𝑥 sub zero, meaning we plug in 𝑥 sub zero into our function. So instead of three 𝑥 minus two, we need three times 𝑥 sub zero minus two.
And we know what 𝑥 sub zero is; it’s equal to two. So we have three times two minus two. Three times two is six, and six minus two is four. So 𝑥 sub one is equal to four. So now to find 𝑥 sub two, we plug in two for 𝑖. So we will evaluate 𝑓 of 𝑥 sub one. And we can replace 𝑥 sub one with four. Three times four is 12, and 12 minus two is 10. So 𝑥 sub two is equal to 10.
So lastly, to find 𝑥 sub three, what we were asked to find, we plug in three for 𝑖, evaluate 𝑓 of 𝑥 sub two, replace 𝑥 sub two with 10. And three times 10 is 30, and 30 minus two is 28. So 𝑥 sub three is equal to 28.