l.4.2 Inverse images of sets We just showed that a function f 1 A r B \"pushes formrd subsets of A to subsets of B.\" We can also \"pull back subsets of 3\": DEFINITION 6. If?" is a subset of B, we dene f"[T] = {a e A :f(a) E T]. The set f'l[T] is a subset of A and is called the inverse image of T under f or the pie-image of T under f. Just as with images. there are two ways to think about inverse images: ' One interpretation involves a test condition on elements ofthe domain A. To nd f'\"l [T], we check each element a of A. We compute f (a) and check whether it is an element of T. If it is, we include it in f\" [T]. If it's not, we don't. - Here is a second interpretation. To compute f"[T], we run through each element 1 e T. We compute [a e A : a) : t}. (This is exactly f"[l!l], the inverse image of a set containing a single point.) We collect the Inverse images; the collection of all elements obtained by this process (running through all elements I e T) is exactly f "[T]. Symbolieally, we have that f \"[T] = Ure'r f'llllll- 1. Carefully explain how you can find f-1[(5, 4, 0, -1}] and f-'[[4, 9]] from the graph of f (x) = x2, where the domain of f is taken to be R. (This exercise should help reinforce what you learned in Your Turns 12 and 13.)