Question

question 6, 6.5.47
part 2 of 4
hw score: 62.5%, 5 of 8 points
points: 0 of 1
find (f ∘ g)(x) and (g ∘ f)(x) and graph each of these functions.
f(x) = -8x
g(x) = cot x
find (f ∘ g)(x).
(f ∘ g)(x) = -8 cot x
choose the correct graph of (f ∘ g)(x).
○ a.
graph a
○ b.
graph b
○ c.
graph c
○ d.
graph d

Explanation:

Step1: Identify composite function

We know $(f \circ g)(x) = f(g(x)) = -8\cot x$.

Step2: Analyze $\cot x$ transformation

The parent function $\cot x$ has vertical asymptotes at $x = n\pi$ (where $n$ is any integer), and it is reflected over the x-axis and vertically stretched by a factor of 8 to get $-8\cot x$. This means the range becomes $(-\infty, \infty)$ with values scaled by 8, and the sign is flipped: where $\cot x$ is positive, $-8\cot x$ is negative, and vice versa.

Step3: Match to graph

Graph A has a y-axis range of $\pm16$, vertical asymptotes at $x=n\pi$, and the correct sign reversal/stretch: when approaching $0^+$, $\cot x \to +\infty$ so $-8\cot x \to -\infty$, which matches the downward trend near $0^+$ in graph A.

Answer:

$(f \circ g)(x) = -8\cot x$
Correct graph: A.