Question

#10. Define f: R ® R defined by f(x) = 4x². Let A = [0, 4] and B = [-3, 0].

#10(a) Find the sets f(A), f(B), f(A) Ç f(B), and f(A Ç B).               Is f(A Ç B) = f(A) Ç f(B)?

#10(b) Find the sets which are inverse images:

            f ^-1(A), f ^-1 (B), f ^-1 (A) Ç f ^-1 (B), and f ^-1 (A Ç B).      Is f ^-1 (A Ç B) = f ^-1 (A) Ç f ^-1 (B)?

#10. Define f: R ® R defined by f(x) = 4x². Let A = [0, 4] and B = [-3, 0].

#10(a) Find the sets f(A), f(B), f(A) Ç f(B), and f(A Ç B).               Is f(A Ç B) = f(A) Ç f(B)?

10(b) Find the sets which are inverse images:

            f ^-1(A), f ^-1 (B), f ^-1 (A) Ç f ^-1 (B), and f ^-1 (A Ç B).      Is f ^-1 (A Ç B) = f ^-1 (A) Ç f ^-1 (B)?

4. Find an example of a relation on a set S that is symmetric , but not reflexive and not transitive. (Note that you can just specify your set S and list an appropriate set of ordered pairs.)

#6. State the range of each function f: R ® R. (no work required to be shown)

      #6(a)    f(x) = 5 sin (4x)

      #6(b)   f(x) = −(x − 3)² + 8

Answer 1

4. Consider the relation R on the set Z, the set of all integers,given T y1. We know that symmetric but it is neither reflexive nor transitive - x0 and example, consider x 1,y 2,z3.) 6.(a). Consider f : RR given by f(x)5.sin(4) R, - y ly - r. Hence R is by Ry (Note that y1 and ly - 1l- 2 1. For sin(4x) takes all real numbers in -1, 1 and varies over As f (x) no numbers lying outside that interval. Hence the range of set -5, 5 6.(b). Consider f: R- As r is the R given by f(x)( -2)2 8 R,-( 2)2 takes all real numbers in o0, 0 and varies over (-00+8,0 8 no other numbers. Hence the range of f() is the set (o0, 8]