Descrete math help Answers for all would be helpful thankyou

Descrete math help. Answers for all would be helpful thankyou

Let a function f be f : R arrow R such that f(x) = 0 when 0. f is plotted in the figure below. a) Determine whether f is one-to-one. b) Determine whether f is onto. c) Determine whether f is total. d) Determine ranges for x and y so that f is total but not one-to-one. e) Determine ranges for x and y so that f is one-to-one, onto and total. (Give proofs to justify all your answers)

Solution

a) f is not one to one as from the graph it is seen that there are two x intercepts.

Hence 2 xs have the same image. Not one to on.

b) Not onto because the graph shows that near 1 and near -2 y do not have preimage in x.

c) f is total because vertical line through any point shows only one intersection of the curve.

d) Ranges for x and y that f is total but not one to one is 0<x<3.2 or3.6<x<6 and -0.6<y<0.6 or -1.4<y<-0.6

In other words x belongs to (0,3.2)U(3.6,6) and y belongs to (-0.6,0.6) U (-1.4,-0.6)

e) Consider -2<x<0 and y =0.

In this range x is onto but not total as y is the same for all x.

Consider the graph for x belonging to (3.2,3.6) and y belongs to (-0.4,-0.6)

In this f is one to one, on to and total as we see that

a) f is one to one as this curve monotonically decreasing with unique value of y for each x.

b) f is onto as the interval from -0.4 to -0.6 each y has a preimage x

c) f is total as vertical test gives one intersection.

Hence proved.