I need full solution for this question.

1. Given two functions f and g, we say that g is a quasi-inverse of f when

" there exists a non-empty, open interval I contained in the domain of f, such that the restriction of f to I is one-to-one, and g is the inverse of that restriction."

for example, arctan is a quasi-inverse of tan.

construct a function f that satisfies all the following properties at once:

(a) The domain of f is R.

(b) f is differentiable

(c) For every c >0 there exists a quasi-inverse g of f such that g is differentiable at 0 and 0 < g' (0) < c .

We want an explicit equation for f and a graph (feel free to use Desmos, for example). Once you have the function, you may use the graph to help justify why it satisfies property (c), rather than proving it entirely algebraically.

use your judgement to decide how much of an explanation you need.