Question

find the domain of the function y = 5/3 tan(3/4 x). a) all real numbers except 0 and odd integer multiples of 4π/3 b) all real numbers except odd integer multiples of 4π/3 c) all real numbers except 0 and odd integer multiples of 2π/3 d) all real numbers except odd integer multiples of 2π/3 question 7 (5 points) which point is an x - intercept of the function y = 2 cot(1/4 x)? a) (0, 0) b) (π, 0) c) (2π, 0)

Explanation:

Step1: Recall tangent - function domain

The tangent function $y = \tan(u)$ is undefined when $u=(2n + 1)\frac{\pi}{2}$, where $n\in\mathbb{Z}$. For the function $y=\frac{5}{3}\tan(\frac{3}{4}x)$, we set $\frac{3}{4}x=(2n + 1)\frac{\pi}{2}$.

Step2: Solve for x

Cross - multiply to get $3x = 2(2n + 1)\frac{\pi}{2}\times4$. Then $3x=(2n + 1)4\pi$, and $x=\frac{4\pi(2n + 1)}{3}$, where $n\in\mathbb{Z}$. This means the function $y=\frac{5}{3}\tan(\frac{3}{4}x)$ is undefined at odd integer multiples of $\frac{4\pi}{3}$. The domain is all real numbers except odd integer multiples of $\frac{4\pi}{3}$.

for second sub - question:

Step1: Recall cotangent - function x - intercept

The $x$ - intercepts of the cotangent function $y = \cot(u)$ occur when $\cot(u)=0$. Since $\cot(u)=\frac{\cos(u)}{\sin(u)}$, $\cot(u) = 0$ when $\cos(u)=0$ and $\sin(u)
eq0$. For the function $y = 2\cot(\frac{1}{4}x)$, we set $\frac{1}{4}x=(2n + 1)\frac{\pi}{2}$, $n\in\mathbb{Z}$ (to find where $\cot$ is zero).

Step2: Solve for x

Multiply both sides by 4: $x = 2(2n+1)\pi$. When $n = 0$, $x = 2\pi$.

Answer:

B. All real numbers except odd integer multiples of $\frac{4\pi}{3}$