Webwork 10 Logarithms Question 1

I don’t necessarily see how f(x)^{\prime }=1+2x+3x^2 was expanded/simplified to result in f(x)^{\prime }=\left(1+x\right)^2+2x^2.

Following that, I do not understand what it means for ‘f’ to be monotone, and I am unsure how to approach finding \left(f^{-1}\right)(40). Do I first need to find \left(f^{-1}\right)? And if so, how do I do that?

A function is monotone if it is always increasing or always decreasing. Furthermore, there’s a nice relationship between monotonicity and the derivative:

• The function f is increasing on an interval if f'(x)>0 for all x in that interval and
• The function f is decreasing on an interval if f'(x)<0 for all x in that interval.

Now, the reason this is important for this particular problem is simply because you are trying to work with the inverse of a function in a fairly complicated situation. You can’t use algebra to find an algebraic expression for the inverse. Nonetheless, you know the inverse exists simply because the function is monotone and, therefore, invertible.

The specifics of converting 1+2x+3x^2 to (1+x)^2+2x^2 is not particularly important. It is pretty easy, though, if you go in the other direction - just expand the second expression and I think you’ll see that you get the first.

The point behind the observation is that the expression with the squares is manifestly positive. Thus, f is always increasing since its derivative is always positive.