a) The number under the square root is either zero or positive
b) The denominator of the fraction is not zero.
Looking at the equation for m(v) we see that to make the number under the square root positive. (It can't be equal to c because that would make the denominator 0.) Thus the domain is [0, c).
(NOTE: I am using the convention that v is a scalar, ie the speed, not the magnitude, so it can't be less than 0.)
As to the range, we can see that m(v) is the smallest for v = 0. We can also see that . So the range of m(v) is .
2) I already used this in the answer to 1), but specifically:
Vertical asymptotes: These occur when the denominator is 0, which I mentioned in 1) only happens when v = c.
Horizontal asymptotes: These occur when . As v can't go to either limit, there is no horizontal asymptote.
4) We wish to investigate the nature of the function, so we take the first and second derivatives:
First look at m'(v): The only place it is 0 is at v = 0. From there the slope is always positive. Thus m(v) is monotonically increasing.
Now look at m''(v): Does this ever equal 0? No. So there are no inflection points. And as m''(v) is positive on [tex] [0, c) the curvature is always positive. Thus m(v) is concave up on its domain.