Find out the range of $x^{\frac{1}{3}}+\sin 2x$
Since the range of the function $\sin2x$ is $[-1,1]$ and the range of the function $x^{\frac{1}{3}}$ is $R$.I found the range of the $x^{\frac{1}{3}}+\sin 2x$ as intersection of the ranges of the two functions $\sin2x$ and $x^{\frac{1}{3}}$ and i found the range as $[-1,1]$ but the range given in my book is $R$.
Just as the domain of sum/difference/product/quotient of two functions is the intersection of the domains of the individual functions.In the same way,i thought that the range of sum/difference/product/quotient of two functions is the intersection of the ranges of the individual functions.
But i am wrong.Is the range of sum/difference/product/quotient of two functions is the union/intersection of the ranges of the individual functions?
Or there is no such ''union/intersection'' relationship exist?Please explain.Thanks.