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In general for a function to have an inverse dierent

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Unformatted text preview: f . From this perspective, we see g ◦ f as a two step process taking an input x and first applying the procedure f then applying the procedure g . Abstractly, we have g f x f (x) g (f (x)) g◦f In the expression g (f (x)), the function f is often called the ‘inside’ function while g is often called the ‘outside’ function. There are two ways to go about evaluating composite functions - ‘inside out’ and ‘outside in’ - depending on which function we replace with its formula first. Both ways are demonstrated in the following example. Example 5.1.1. Let f (x) = x2 − 4x, g (x) = 2 − √ x + 3, and h(x) = indicated composite functions. State the domain of each. 1. (g ◦ f )(x) 2x . Find and simplify the x+1 5. (h ◦ h)(x) 2. (f ◦ g )(x) 6. (h ◦ (g ◦ f ))(x) 3. (g ◦ h)(x) 4. (h ◦ g )(x) 7. ((h ◦ g ) ◦ f )(x) Solution. 1. By definition, (g ◦ f )(x) = g (f (x)). We now illustrate the two ways to evaluate this. • inside out : We insert the expression f (x) into g first to get (g ◦ f )(x)...
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