Functions: Introduction to functions
The range of a function
The range
Consider the real function #f#.
The set of all values #f(x)# for #x# in the domain of #f# is called the range of #f#.
If #p# is a real number in the domain of #f# with #f(p)=0#, then #p# is called a zero of #f#.
If #f# has a zero, then #0# is part of the range of #f#.
In general, a point #y# lies in the range of #f# if and only if the equation #f(x)=y# with unknown #x# has a solution. The range of #f# is therefore the set of all possible values #y# for which the equation #y=f(x)# with unknown #x# from the domain of #f# has a solution.
The range of a function depends on the domain of the function: the larger we choose the domain, the larger in general the range will be.
#0#
Indeed,
Indeed,
- the equation #0={{8}\over{x-9}}# with unknown #x# has no real solution, and,
- for every #y\ne0#, the equation #y={{8}\over{x-9}}# has a solution in #x#, namely #x={{8}\over{y}}+9#.
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