Question #7d077

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Answer 1 · 2018-01-11T17:27:27

See below.

A function is one to one if:

No two elements in the domain map on to the same element in the range.

Since we have a restricted domain, this criteria is met, so #f(x)# is one to one

For an unrestricted domain #f(x)=x^2# is a many to one function.

Inverse of #f(x)#

We need to express #x# as a function of #y#:

#y=x^2#

#x=+-sqrt(y)#

Substituting #x=y#

#y=+-sqrt(x)# #:.# #f^-1(x)=+-sqrt(x)#

Domain of #f(x)#

For #x<=0#

Domain is #{x in RR: -oo < x<=0}#

Since #x^2>=0# for all #RR#

Range is:

#{y in RR:0 <=y< oo}#

Domain of #f^-1(x)#

Because the domain of #f(x)# is #x<=0#, only the inverse #y=-sqrt(x)# is needed. The domain of this will be the same as the range of #f(x)# i.e.

#{x in RR: 0 <= x< oo}#

The range will be the same as the domain of #f(x)# i.e.

#{y in RR : -oo < y <= 0 }#

For one to one functions the range of #f(x)# is the domain of #f^-1(x)# and the domain of #f(x)# is the range of #f^-1(x)#.