Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 5 - Exponential and Logarithmic Functions - 5.1 Composite Functions - 5.1 Assess Your Understanding - Page 255: 27


a) $x^4+8x^2+16$; $D=(-\infty,\infty)$ b) $x^4+4$; $D=(-\infty,\infty)$ c) $x^4$; $D=(-\infty,\infty)$ d) $x^4+8x^2+20$; $D=(-\infty,\infty)$

Work Step by Step

We are given the functions: $f(x)=x^2$ $g(x)=x^2+4$ Let's note: $D_f$=the domain of $f$ $D_g$=the domain of $g$ We have: $D_f=(-\infty,\infty)$ $D_g=(-\infty,\infty)$ a) Determine $f\circ g$ and its domain $D_{f\circ g}$: $(f\circ g)(x)=f(g(x))=f(x^2+4)=(x^2+4)^2=x^4+8x^2+16$ $D_{f\circ g}=(-\infty,\infty)$ b) Determine $g\circ f$ and its domain $D_{g\circ f}$: $(g\circ f)(x)=g(f(x))=g(x^2)=(x^2)^2+4=x^4+4$ $D_{g\circ f}=(-\infty,\infty)$ c) Determine $f\circ f$ and its domain $D_{f\circ f}$: $(f\circ f)(x)=f(f(x))=f(x^2)=(x^2)^2=x^4$ $D_{f\circ f}=(-\infty,\infty)$ d) Determine $g\circ g$ and its domain $D_{g\circ g}$: $(g\circ g)(x)=g(g(x))=g(x^2+4)=(x^2+4)^2+4=x^4+8x^2+20$
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