0
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

No there is no direct "union/intersection" relationship.

Consider two functions with ranges that are a single interval, $[a,b]$ and $[c,d]$. Clearly, the sum of two values taken from these ranges cannot exceed $b+d$ nor be smaller than $a+c$, but there is no guarantee that these bounds can be reached, as the two function values do not vary independently.

As an extreme case, consider $f(x)=\sin(x)$ and $g(x)=-\sin(x)$, both with range $[-1,1]$. But $f(x)+g(x)=0$ !

All you can say a priori is that $\text{range}(f+g)\subseteq[a+c,b+d]$. More generally, $\text{range}(f+g)\subseteq\text{range}(f)\oplus\text{range}(g)$, where $\oplus$ denotes the Minkowski sum of the two sets.

To get the exact range, you need to perform a complete study, where knowing the ranges of the original functions helps little.

Similarly, $\text{range}(f-g)\subseteq[a-c,b-d]$ or $\subseteq\text{range}(f)\oplus\text{range}(-g)$. The cases of the product and quotient are a little more tricky...

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ In the last line, why isn't the range given by $\text{range}(f-g) \subseteq [a - d, b - c ]$ ? $\endgroup$ – DWade64 Jul 31 '18 at 14:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.