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 (non-identity)
functions. Check your answer by performing the function composition.
1. F (x) = |3x − 1|
2. G(x) = x2 2
3. H (x) = √
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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