hey guys, could you please help with this - i have been seacrhing the inernet and textbooks adn have not found an answer yet, and this is urgent.
we are currently doing functions in 1st year university, and often in order to justify why a function has an inverse, we must restrict the domain to the largest interval for which the function is strictly increasing (or decreasing).
now, my lecturer defined 'strictly increasing' as 'if x1 < x2, f(x1) < f(x2)'.
now in order to find where a function is strictly increasing, i find the derivative and find for what domain it is greater than 0 for.
however, my lecturer gave us an example, saying for y=sinx, it is strictly increasing over [-pi/2, pi/2] ---> this is a CLOSED INTERVAL
So as i do problems now, i am having trouble identifying whether the stationary points should be included in the largest interval for which f(x) is strictly increasing (as my lecturer said for sinx), or not
ie. if a question asks to find over what interval a function is strictly increasing, do we include the stationary points (ie make it a closed interval, beacuse technically x1 < x2, f(x1) < f(x2) still holds, however, if finding it by finding when f'(x)>0, you dont get those points???
could someone clarify this further by telling me the largest interval in which this function is strictly increasing : f(x) = x/(1+x^2)
is it (-1,1) or [-1,1]