We will revisit this concept again in chapter 8 3 296

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Unformatted text preview: ◦ f )(x) = g (f (x)), provided x is an element of the domain of f and f (x) is an element of the domain of g . 1 These were introduced, as you may recall, as piecewise-defined linear functions. This is a really bad math pun. 3 If we broaden our concept of functions to allow for complex valued coefficients, the Complex Factorization Theorem, Theorem 3.14, tells us every function we have studied thus far is a combination of linear functions. 2 280 Further Topics in Functions The quantity g ◦ f is also read ‘g composed with f ’ or, more simply ‘g of f .’ At its most basic level, Definition 5.1 tells us to obtain the formula for (g ◦ f ) (x), we replace every occurrence of x in the formula for g (x) with the formula we have for f (x). If we take a step back and look at this from a procedural, ‘inputs and outputs’ perspective, Defintion 5.1 tells us the output from g ◦ f is found by taking the output from f , f (x), and then making that the input to g . The result, g (f (x)), is the output from g ...
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