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
