a) Show that for every x, there exists only exactly one y.
b) Let the solution in (a) be Show that this function is locally.
c) With the previous two answers in mind, how does it now follow that
d) What is the domain and range of f.
Let and . Then g and h are both infinitely differentiable functions. Also, h is strictly increasing, with domain and range both equal to R, so by the inverse function theorem h has a continuously differentiable inverse . The range of g is the half line [10,∞), with the minimum value 10 occurring at x=0.
If h(y) = g(x) then , which by the chain rule is a continuously differentiable function of x. The domain of this function is the whole of R, and its range is the half line .
[I can't help wondering whether the person who set this question was thinking that g(0)=9 rather than 10. In that case, since h(2)=9, the answer to the last part of the question would come out much more cleanly: the range of f would then be the half line [2,∞).]