Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 1 - Limits and Continuity - 1.6 Continuity of Trigonometric Functions - Exercises Set 1.6 - Page 105: 9

Answer

a-c. Continuous

Work Step by Step

Theorem 1.5.6(b) states that if $f(x)$ and $g(x)$ are continuous, then $f \circ g$ is also continuous. a. Let $f(x) = sin(x)$. Let $g(x) = x^3 + 7x + 1$. Note that both $f(x)$ and $g(x)$ are continuous as the sine function is continuous and all polynomials are continuous. Thus, by Theorem 1.5.6(b), $f \circ g = sin(x^3+7x+1)$ is also continuous. b. Let $f(x) = |x|$. Let $g(x) = sin(x)$. Note that both $f(x)$ and $g(x)$ are continuous as an absolute value function and sine function, respectively. Thus, by Theorem 1.5.6(b), $f \circ g = |sin(x)|$ is also continuous. c. Let $f(x) = x^3$. Let $g(x) = cos(x)$. Let $h(x) = x+1$. Note that both $f(x)$, $g(x)$, and $h(x)$ are continuous as a polynomial, cosine function, and polynomial, respectively. Thus, by Theorem 1.5.6(b), $g \circ h = cos(x+1) $ is also continuous. Thus, by Theorem 1.5.6(b), $f \circ (g \circ h) = cos^3(x+1)$ is also continuous.
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