Unformatted text preview: hich our last example illustrates.
Example 5.1.3. Write each of the following functions as a composition of two or more (nonidentity)
functions. Check your answer by performing the function composition.
1. F (x) = 3x − 1
2. G(x) = x2 2
+1 √
x+1
3. H (x) = √
x−1
Solution. There are many approaches to this kind of problem, and we showcase a diﬀerent
methodology in each of the solutions below.
1. Our goal is to express the function F as F = g ◦ f for functions g and f . From Deﬁnition
5.1, we know F (x) = g (f (x)), and we can think of f (x) as being the ‘inside’ function and g
as being the ‘outside’ function. Looking at F (x) = 3x − 1 from an ‘inside versus outside’
perspective, we can think of 3x − 1 being inside the absolute value symbols. Taking this
cue, we deﬁne f (x) = 3x − 1. At this point, we have F (x) = f (x). What is the outside
function? The function which takes the absolute value of its input, g (x) = x. Sure enough,
(g ◦ f )(x) = g (f (x)) = f (x) = 3x − 1 = F (x), so we are done. 288 Further Topics in Functions 2. We at...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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