Question

Find the domain and range of $5\sin \left(x-\frac{\mathrm{\pi }}{6}\right)$

Answer 1

Solve for the domain and range:

Let $y=5\sin \left(x-\frac{\mathrm{\pi }}{6}\right)$

The domain of the expression is all real numbers except where the expression is undefined. For the function $y=5\sin \left(x-\frac{\mathrm{\pi }}{6}\right)$ there is no value of $x$ for which the function is not defined. So domain : $(-\infty ,\infty )$

The lower bound of the range for $5\sin \left(x-\frac{\mathrm{\pi }}{6}\right)$ is found by substituting the negative magnitude of the coefficient into the equation and we get $y=$$-5$

The upper bound of the range for $5\sin \left(x-\frac{\mathrm{\pi }}{6}\right)$ is found by substituting the positive magnitude of the coefficient into the equation and we get $y=$$5$

Range :$[-5,5],\{-5\le y\le 5\}$

Hence, domain is $(-\infty ,\infty ),\{x\in R\}$ and range is $[-5,5],\{-5\le y\le 5\}$