Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 1 - Functions and Limits - 1.3 New Functions from Old Functions - 1.3 Exercises: 61

Answer

Yes, $f \circ g$ is linear. The slope of $f \circ g$ is $m_1m_2$.

Work Step by Step

$$(f \circ g)(x)= f(g(x))=m_1(m_2x++b_2)+b_1$$ $$=m_1m_2x+m_1b_2+b_1$$The composite function $f \circ g$ is linear since it is of the form $mx+b$, with $m=m_1m_2$ and $b=m_1b_2+b_1$. The slope of the linear function $mx+b$ is $m$, so the slope of the composite function $f \circ g$ is $m_1m_2$.
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