College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 6 - Section 6.1 - Composite Functions - 6.1 Assess Your Understanding - Page 409: 27

Answer

$ a.\qquad (f\circ g)(x)=x^{4}+8x^{2}+16,\qquad$ Domain: $\mathbb{R}$ $ b.\qquad (g\circ f)(x)=x^{4}+4,\qquad$ Domain: $\mathbb{R}$ $ c.\qquad (f\circ f)(x)=x^{4},\qquad$ Domain: $\mathbb{R}$ $ d.\qquad(g\circ g)(x)=x^{4}+8x^{2}+20,\qquad$ Domain: $\mathbb{R}$

Work Step by Step

$ f(x)=x^{2},\qquad$ Domain: $\mathbb{R}$ $ g(x)=x^{2}+4,\qquad$ Domain: $\mathbb{R}$ $a.$ $f\circ g(x)=f[g(x)]=[g(x)]^{2}$ $=(x^{2}+4)^{2}$ $=x^{4}+8x^{2}+16,\qquad$ Domain: $\mathbb{R}$ $b.$ $(g\circ f)(x)=g[f(x)] =[f(x)]^{2}+4$ $=(x^{2})^{2}+4$ $=x^{4}+4,\qquad$ Domain: $\mathbb{R}$ $c.$ $f\circ f(x)=f[f(x)]=[f(x)]^{2}$ $=(x^{2})^{2}$ $=x^{4},\qquad$ Domain: $\mathbb{R}$ $d.$ $(g\circ g)(x)=g[g(x)] =[g(x)]^{2}+4$ $=(x^{2}+4)^{2}+4$ $=x^{4}+8x^{2}+16+4$ $=x^{4}+8x^{2}+20,\qquad$ Domain: $\mathbb{R}$
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