Stitz-Zeager_College_Algebra_e-book

Lets assume that the cobb douglas production model

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Unformatted text preview: ll run into the same trouble as before, but when we check the composition, the domain restriction on g saves the day. We get g −1 ◦ g (x) = √ g −1 (g (x)) = g −1 x2 = x2 = |x| = x, since x ≥ 0. Checking g ◦ g −1 (x) = g g −1 (x) = √ √2 g ( x) = ( x) = x. Graphing6 g and g −1 on the same set of axes shows that they are reflections about the line y = x. y y = g (x) 8 y=x 7 6 5 y = g −1 (x) 4 3 2 1 1 2 3 4 5 6 7 8 x Our next example continues the theme of domain restriction. Example 5.2.3. Graph the following functions to show they are one-to-one and find their inverses. Check your answers analytically using function composition and graphically. 1. j (x) = x2 − 2x + 4, x ≤ 1. 2. k (x) = √ x+2−1 Solution. 1. The function j is a restriction of the function h from Example 5.2.1. Since the domain of j is restricted to x ≤ 1, we are selecting only the ‘left half’ of the parabola. We see that the graph of j passes the Horizontal Line Test and thus j is invertible. y 6 5 4 3 2 1 1 2 −1 y = j (x) √ 6 We graphed y = x in Section 1.8. x 5.2 Inverse Functions 30...
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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