Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 1 - Section 1.3 - New Functions from Old Functions - 1.3 Exercises - Page 44: 33

Answer

(a) $f(g(x)) = 3x^2+3x+5$; Domain $(-\infty, \infty)$ (b) $g(f(x)) = 3(3x^2+11x+10)$; Domain $(-\infty, \infty)$ (c) $f(f(x)) = 9x+20$; Domain $(-\infty, \infty)$ (d) $g(g(x)) = x(x^3+2x^2+2x+1)$; Domain $(-\infty, \infty)$

Work Step by Step

$f(x) = 3x+5$ $g(x)= x^2+x$ (a) $f(g(x)) = 3(x^2+x)+5 = 3x^2+3x+5$ Since we have no restrictions, the domain is R, $(-\infty, \infty)$. (b) $g(f(x)) = (3x+5)^2+3x+5 = 9x^2+30x+25+3x+5 = 9x^2+33x+30 = 3(3x^2+11x+10)$ Since we have no restrictions, the domain is R, $(-\infty, \infty)$. (c) $f(f(x)) = 3(3x+5)+5 = 9x+15+5 = 9x+20$ Since we have no restrictions, the domain is R, $(-\infty, \infty)$. (d) $g(g(x)) = (x^2+x)^2+x^2+x = x^4+2x^3+x^2+x^2+x = x^4+2x^3+2x^2+x = x(x^3+2x^2+2x+1)$ Since we have no restrictions, the domain is R, $(-\infty, \infty)$.
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