Determine whether each of the functions below is onto andor

Determine whether each of the functions below is onto, and/or one-to-one, prove your answers.

Determine whether each of the functions below is onto, and/or one-to-one, prove your answers. (a) f: Z rightarrow Z, f(x) = x - 1 (b) f: R rightarrow R, f(x) = -3x^2 + 7 (c) f: Z rightarrow Z, f(x) = [x] (d) f: R rightarrow R, f(x) = x (e) f: R rightarrow z, f(x) = x

Solution

a) this is onto function as for evey value of y we have a value of x

b)this is not 1 to 1 as for negative value of x we get the same value of y as that for positive value of x.

c)this is onto function as for evey value of y we have a value of x

d)this is onto function as for evey value of y we have a value of x

e) this is neither of the above as we do not have the value of x for every y as here x is integer where y is a real number