Please answer in detail.

3.19 Let C([ - 1, 1]) be the vector space of continuous functions on the interval [ - 1, 1] and define T : C([-1, 1]) - R by TO) = [ f(x)de Thus, for example, a. Let f(x) = et and g(x) = x2 for - 1 s x s 1. Show by direct computation that T(2f + 3g) = 2T(f) + 3T(g) b. Prove that T is a linear transformation. c. Let S be the transformation defined by S(n) = / (f( x)2 dx Show by explicit computation that S(2e) + 28(e*). Is S linear? d. For the transformation S in part (c), find two functions fand g, f + 9, such that S(f + 9) # S(f) + S(g) e. Let U be defined by U(f) = f(x)x dx Is U linear? Prove your