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Numerade Educator

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Problem 62 Hard Difficulty

If you invest $ x $ dollars at 4% interest compounded annually, then the amount $ A(x) $ of the investment after one year is $ A(x) = 1.04x $. Find $ A \circ A $, $ A \circ A \circ A $, and $ A \circ A \circ A \circ A $. What do these compositions represent? Find a formula for the composition of $ n $ copies of $ A $.

Answer

$1.04^{n} x$

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Video Transcript

all right. We start with a of X equals 1.0 for X, and then we find a of A so that would be 1.4 times 1.0 for X. And that's the same as 1.0 for square times X. Now we find a of a of a So what we can do we already know a of a Let's do one more time Put a inside that when we get 1.4 squared times 1.4 x So all I did there was I put the original inside, the one we just did, and we can simplify that and call it 1.4 Cubed X and then for a of a of a of A we're going to take what we just got and put one more a inside it, and that will be 1.4 cubed times 1.4 x. And that's going to be 1.0, for to the fourth times x So each of these represent the amount of money in the investment after another year has gone by. So this is one year, two years, three years four years. Now, How about in years if you multiply or if you compose this function end times, it's not multiplying. The function is composing the function. End times you'll end up with following the pattern 1.4 to the end. Power times X.