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Unformatted text preview: output from f , f (x) = 3x + 4, and
substitute that into g . That is, g (f (x)) = g (3x + 4) = (3x+4)−4 = 33 = x, which is our original
input to f . If we carefully examine the arithmetic as we simplify g (f (x)), we actually see g ﬁrst
‘undoing’ the addition of 4, and then ‘undoing’ the multiplication by 3. Not only does g undo
f , but f also undoes g . That is, if we take the output from g , g (x) = x−4 , and put that into
f , we get f (g (x)) = f x−4 = 3 x−4 + 4 = (x − 4) + 4 = x. Using the language of function
composition developed in Section 5.1, the statements g (f (x)) = x and f (g (x)) = x can be written
as (g ◦ f )(x) = x and (f ◦ g )(x) = x, respectively. Abstractly, we can visualize the relationship
between f and g in the diagram below.
f x = g (f (x)) y = f (x) g 294 Further Topics in Functions The main idea to get from the diagram is that g takes the outputs from f and returns them to
their respective inputs, and conversely, f takes outputs...
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