Problem 4. Prove that if X is path connected and f: X→Y is continuous, then im(f) is path connected.

8. Prove or disprove the following: (a) If X and Y are path-connected, then X x Y is path-connected. (b) If A C X is path-connected, then A is path-connected. (c) If X is locally path-connected, and AC X, then A is locally path-connected. (d) If X is path-connected, and f: X Y is continuous, then f(X) is path- connected. (e) If X is locally path-connected, and f: X→ Y is continuous, then f(X) is locally path-connected.

6. Determine whether each of the following functions is one-to-one, onto, neither or both. (a) f : {0, 1}² → {0, 1}³. ƒ(x) = x0, i.e., append a '0' to the end of string 'x'. (b) ƒ : R → R, f(x) = –3x +4 (c) f : R → R, f() = [x/5]

1. Iƒ ƒ(x) = −3x +4 and g(x) = x², find (gof)(-2) -8 4 -100 100 * REQUIRED 1

9 c and d