3. given y= 2 tan (3x - π/6) + 7, determine its characteristics such as: a) domain b) range c) amplitude d)…

Explanation:

Step1: Find domain

The tangent function $y = \tan(u)$ is undefined when $u=(n +\frac{1}{2})\pi$, where $n\in\mathbb{Z}$. Let $u = 3x-\frac{\pi}{6}$. Then $3x-\frac{\pi}{6}\neq(n+\frac{1}{2})\pi$. Solving for $x$ gives $x\neq\frac{(n+\frac{1}{2})\pi+\frac{\pi}{6}}{3}=\frac{(3n + 2)\pi}{9},n\in\mathbb{Z}$. So the domain is $\left{x\in\mathbb{R}\mid x\neq\frac{(3n + 2)\pi}{9},n\in\mathbb{Z}\right}$.

Step2: Find range

The range of the tangent - function $y = \tan(u)$ is $(-\infty,\infty)$. For $y = 2\tan(3x-\frac{\pi}{6})+7$, the range is also $(-\infty,\infty)$ since multiplying by 2 and adding 7 does not change the range of the tangent function.

Step3: Find amplitude

The tangent function $y = A\tan(Bx - C)+D$ has no amplitude because the tangent function is unbounded. So the amplitude is undefined.

Step4: Find period

The period of the tangent function $y=\tan(Bx - C)$ is given by $T=\frac{\pi}{|B|}$. Here $B = 3$, so the period $T=\frac{\pi}{3}$.

Step5: Find phase shift

For the function $y = A\tan(Bx - C)+D$, the phase - shift is given by $\frac{C}{B}$. Here $C=\frac{\pi}{6}$ and $B = 3$, so the phase shift is $\frac{\frac{\pi}{6}}{3}=\frac{\pi}{18}$.

Step6: Find vertical shift

For the function $y = A\tan(Bx - C)+D$, the vertical shift is $D$. Here $D = 7$, so the vertical shift is 7.

Answer:

a) Domain: $\left{x\in\mathbb{R}\mid x\neq\frac{(3n + 2)\pi}{9},n\in\mathbb{Z}\right}$ b) Range: $(-\infty,\infty)$ c) Amplitude: Undefined d) Period: $\frac{\pi}{3}$ e) Phase Shift: $\frac{\pi}{18}$ f) Vertical Shift: 7