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Stitz-Zeager_College_Algebra_e-book

# State the restricted domain 2 find and interpret p1

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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 +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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