Question

find (g ∘ f)(x).
(g ∘ f)(x) = cot(-8x)
choose the correct graph of (g ∘ f)(x).
○ a.
○ b.
○ c.
○ d.

Explanation:

Step1: Simplify the cotangent function

Use the identity $\cot(-\theta) = -\cot(\theta)$, so:
$\cot(-8x) = -\cot(8x)$
The graph of $-\cot(8x)$ is the reflection of $\cot(8x)$ over the x-axis, with the same period.

Step2: Calculate the period of the function

The period of $\cot(kx)$ is $\frac{\pi}{|k|}$. For $k=8$:
$\text{Period} = \frac{\pi}{8}$

Step3: Match period to graph options

We need a graph where the distance between consecutive vertical asymptotes (the period) is $\frac{\pi}{8}$. On an x-axis scaled with $\pi$, this means asymptotes are spaced $\frac{\pi}{8}$ apart. Additionally, the negative sign flips the standard cotangent curve (which decreases from left to right between asymptotes) to increase from left to right between asymptotes.

Option D has the correct period $\frac{\pi}{8}$ and the correct orientation from the negative sign.

Answer:

D. (The graph with vertical asymptotes spaced $\frac{\pi}{8}$ apart, where the curve increases between consecutive asymptotes)