Question

Let a function f be f: R rightarrow R such that f(x) = 0 when x lessthanorequalto 0 and f(x) = log(x) * sin(x) when x > 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)

Answer 1

a
f(x) is not one to one because for x < 0, f(x) = 0 i.e. value does not anyore depend on x

b
f(x) is onto function since log(x) is onto in nature and sin(x) is an oscillating. We always get an onto function on multiplying onto function with oscilating function

c
f(x) is a total functon since it is defined for all +ve x

d
for x > 0, the function is total and not one to one

Am sorry buddy, but we are allowed to answer only 1 answer at a time :(