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inside out: We insert the expressionf(x) intoh◦gfirst to get((h◦g)◦f)(x)=(h◦g)(f(x)) = (h◦g)(x2-4x)=4-2p(x2-4x) + 33-p(x2-4x) + 3=4-2√x2-4x+ 33-√x2-4x+ 3outside in: We use the formula for (h◦g)(x) first to get((h◦g)◦f)(x)=(h◦g)(f(x)) =4-2p(f(x)) + 33-pf(x)) + 3=4-2p(x2-4x) + 33-p(x2-4x) + 3=4-2√x2-4x+ 33-√x2-4x+ 3We note that the formula for ((h◦g)◦f)(x) before simplification is identical to that of(h◦(g◦f))(x) before we simplified it.Hence, the two functions have the same domain,h◦(f◦g) is (-∞,2-√10)∪(2-√10,1]∪3,2 +√10)∪(2 +√10,∞).It should be clear from Example5.1.1that, in general, when you compose two functions, such asfandgabove, the order matters.4We found that the functionsf◦gandg◦fwere differentas wereg◦handh◦g.Thinking of functions as processes, this isn’t all that surprising.If wethink of one process as putting on our socks, and the other as putting on our shoes, the order inwhich we do these two tasks does matter.5Also note the importance of finding the domain of thecomposite functionbeforesimplifying.For instance, the domain off◦gis much different thanits simplified formula would indicate. Composing a function with itself, as in the case of finding(g◦g)(6) and (h◦h)(x), may seem odd. Looking at this from a procedural perspective, however,this merely indicates performing a taskhand then doing it again - like setting the washing machineto do a ‘double rinse’. Composing a function with itself is called ‘iterating’ the function, and wecould easily spend an entire course on just that.The last two problems in Example5.1.1serveto demonstrate theassociativeproperty of functions. That is, when composing three (or more)functions, as long as we keep the order the same, it doesn’t matter which two functions we composefirst. This property as well as another important property are listed in the theorem below.4This shows us function composition isn’tcommutative. An example of an operation we perform on two functionswhich is commutative is function addition, which we defined in Section1.5. In other words, the functionsf+gandg+fare always equal. Which of the remaining operations on functions we have discussed are commutative?5A more mathematical example in which the order of two processes matters can be found in Section1.7. In fact,all of the transformations in that section can be viewed in terms of composing functions with linear functions.
5.1 Function Composition405Theorem 5.1. Properties of Function Composition:Supposef,g, andhare functions.h◦(g◦f) = (h◦g)◦f, provided the composite functions are defined.IfIis defined asI(x) =xfor all real numbersx, thenI◦f=f◦I=f.