Question Video: Iterating a Function Mathematics

Given an initial value π‘₯β‚€, we can iterate a function using the recursive formula π‘₯_𝑖 = 𝑓(π‘₯_(𝑖 βˆ’ 1)). Find π‘₯₃ given the function 𝑓(π‘₯) = 3π‘₯ βˆ’ 2 with initial value π‘₯β‚€ = 2.

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Video Transcript

Given an initial value π‘₯ sub zero, we can iterate a function using a recursive formula. π‘₯ sub 𝑖 equals 𝑓 of π‘₯ sub 𝑖 minus one. Find π‘₯ sub three, given the function 𝑓 of π‘₯ equals three π‘₯ minus two with an initial value of π‘₯ sub zero equals two.

So our initial value is π‘₯ sub zero equals two. So if we would like to find π‘₯ 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 π‘₯ sub zero, meaning we plug in π‘₯ sub zero into our function. So instead of three π‘₯ minus two, we need three times π‘₯ sub zero minus two.

And we know what π‘₯ 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 π‘₯ sub one is equal to four. So now to find π‘₯ sub two, we plug in two for 𝑖. So we will evaluate 𝑓 of π‘₯ sub one. And we can replace π‘₯ sub one with four. Three times four is 12, and 12 minus two is 10. So π‘₯ sub two is equal to 10.

So lastly, to find π‘₯ sub three, what we were asked to find, we plug in three for 𝑖, evaluate 𝑓 of π‘₯ sub two, replace π‘₯ sub two with 10. And three times 10 is 30, and 30 minus two is 28. So π‘₯ sub three is equal to 28.

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