Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 1 - Precalculus Review - 1.3 The Basic Classes of Functions - Exercises - Page 23: 27

Answer

$fog=\sqrt {x+1}$ The domain of $fog$: {x: x$\geq$-1} $gof=\sqrt x +1$ The domain of $gof$: {x: x$\geq$0}

Work Step by Step

$fog=f(g(x))=f(x+1)=\sqrt {x+1}$ Domain of g is the set of all reals and that of f is all non-negative reals. This implies that x+1 should be non-negative. $\implies\,\,$ x should be greater than or equal to -1. That is, the domain of $fog$ is the set {x: x$\geq$-1} $gof=g(f(x))=g(\sqrt x)=\sqrt x +1$. As the domain of g is the set of all real numbers, the domain of $gof$ is simply the domain of f and that is {x: x$\geq$0}
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