What are examples of a function which is (a) onto but not one-to-one; (b) one-to-one but not onto, with a domain and range of #(-1,+1)#?

Question

Lily Anderson · Accepted Answer

Examples
onto but not one-to-one: #f(x)=1/(x+2)#
one-to-one but not onto: #g(x)=2abs(x)-1# If #f(x)=1/(x+2)# then #color(white)("XXX")#if #f(x_1)=f(x_2)# #color(white)("XXX")#then (noting that for #x_1, x_2 in (-1,+1)#) #color(white)("XXXXXXX")x_2+2=x_1+2# (after cross multiplying) #color(white)("XXXXXXX")rarr x_2=x_1# #color(white)("XXXXXXX")#which implies #f(x)# one-to-one however #color(white)("XXX")#there is no value #barx# for which #color(white)("XXX")f(barx)=0# (which is a value in the specified Range: #(-1,+1)#) #color(white)("XXX")#which implies #f(x)# is not onto. If #g(x)=2abs(x)-1# then #color(white)("XXX")#any value #g(x) in (-1,+1)# #color(white)("XXX")#can be generated by some value of #x# #color(white)("XXX")#which implies #g(x)# is onto however #color(white)("XXX")#if #x_1=-1/2# and #x_2=+1/2# #color(white)("XXX")#then # g(x_1)=g(x_2)# but #x_1!=x_2# #color(white)("XXX")#which implies #g(x)# is not one-to-one

Madelyn Anderson · Answer

undefined (a) An example of a function that is onto but not one-to-one with a domain and range of (-1, +1) is $ f(x) = x^2 $.

(b) An example of a function that is one-to-one but not onto with a domain and range of (-1, +1) is $ g(x) = \frac{x}{2} $.