math question

1. (30 points) Let f be a function defined by xx- 5 f(x) = x - 3 if r > 0 e"(2x -3) if x 50 whenever it is defined. (a) (2 points) Represent f as a case-defined function over its natural domain without using the absolute value function. (b) (3 points) Determine all the vertical asymptote to the graph of f, if any. (c) (5 points) Determine all the horizontal/oblique asymptote to the graph of f, if any. (d) (8 points) Compute f' whenever it exists. When f is not differentiable at ce dom (f), indicate it and justify your claim. You should decide whether you should apply differentiation rules or theorems, or use the definition of differentiability. (e) (6 points) Determine the critical points of f, and hence determine the intervals of increase as well as the intervals of decrease of f. (f) (6 points) Determine the intervals where f is convex and the intervals where f is concave. Hence determine the inflection points, if any