### 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.

The range of the function #f# defined by #f(x) = {{17}\over{x+9}}# on all real numbers except #-9# (that is, having domain #{\mathbb R}\setminus\{-9\}#) consists of all real numbers, except one. Which one?

Solution #0#

Indeed,

Indeed,

- the equation #0={{17}\over{x+9}}# with unknown #x# has no real solution, and,
- for every #y\ne0#, the equation #y={{17}\over{x+9}}# has a solution in #x#, namely #x={{17}\over{y}}-9#.

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