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.