Question

find the period of the function y = 2/3 cos(4/7x) + 2.
a) 7/2π
b) 3π
c) 7/4π
d) 4/7π
question 20 (5 points)
find the domain of the function y = 2 cot(5/8x).
a) all real numbers except odd integer multiples of 4π/5
b) all real numbers except 0 and integer multiples of 8π/5
c) all real numbers except 0 and integer multiples of 4π/5

Explanation:

Step1: Recall period formula for cosine function

The general form of a cosine function is $y = A\cos(Bx - C)+D$, and its period $T=\frac{2\pi}{|B|}$. For the function $y=\frac{2}{3}\cos(\frac{4}{7}x)+2$, $B = \frac{4}{7}$.

Step2: Calculate the period

Using the formula $T=\frac{2\pi}{|B|}$, substitute $B=\frac{4}{7}$ into it. We get $T=\frac{2\pi}{\frac{4}{7}}=2\pi\times\frac{7}{4}=\frac{7\pi}{2}$.

Step1: Recall the domain of cotangent function

The cotangent function $y = \cot(x)$ is undefined when $x = n\pi$, where $n\in\mathbb{Z}$. For the function $y = 2\cot(\frac{5}{8}x)$, we set $\frac{5}{8}x=n\pi$, where $n\in\mathbb{Z}$.

Step2: Solve for x to find the excluded values

From $\frac{5}{8}x=n\pi$, we can solve for $x$: $x=\frac{8n\pi}{5}$, $n\in\mathbb{Z}$. So the domain is all real - numbers except integer multiples of $\frac{8\pi}{5}$.

Answer:

A. $7/2\pi$