Answer
All 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 $f(x) = |x|$. Let $g(x) = 3+x$. Let $h(x) = sin(x)$. Let $i(x) = 2x$. Note that $f(x)$, $g(x)$, $h(x)$, and $i(x)$ are continuous as a absolute value function, polynomial, sine function, and polynomial, respectively. Thus, by Theorem 1.5.6(b), $h \circ i = sin(2x)$ which is continuous. Thus, by Theorem 1.5.6(b), $g \circ (h \circ i) = 3+sin(2x)$ is also continuous. Thus, by Theorem 1.5.6(b), $f \circ (g \circ (h\circ i)) = |3+sin(2x)|$ is also continuous.
b. Let $f(x) = sin(x)$. Let $g(x) = sin(x)$. Note that both $f(x)$ and $g(x)$ are continuous as sine functions. Thus, by Theorem 1.5.6(b), $f \circ g = sin(sin(x))$ is also continuous.
c. Let $f(x) = x^5 - 2x^3 + 1$. Let $g(x) = cos(x)$. Note that both $f(x)$ and $g(x)$ are continuous as a polynomial and cosine function, respectively. Thus, by Theorem 1.5.6(b), $f \circ g = cos^5(x)-2cos^3(x)+1$ is also continuous.
