Reduce a fractionGiven the function f ( x ) = x 2 − 5 x + 5 If I draw this function in maple, I will get a line. How can that be true? I should expect a line except in area of x = − 5 , where f ( x ) → ∞ or f ( x ) → − ∞. Of course Maple has factorized the numerator and reduced. x 2 − 5 x + 5 = x − 5 , x ≠ − 5 My question is now, how can we ever reduce such a fraction with only condition x ≠ − 5 , when it is " − 5 and around it".

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Reduce a fractionGiven the function  f  (  x  )  =                    x        2            −      5              x      +              5            If I draw this function in maple, I will get a line. How can that be true? I should expect a line except in area of   x  =  −      5  , where   f  (  x  )  →  ∞ or   f  (  x  )  →  −  ∞. Of course Maple has factorized the numerator and reduced.                    x        2            −      5              x      +              5              =  x  −      5    ,     x  ≠  −      5  My question is now, how can we ever reduce such a fraction with only condition   x  ≠  −      5  , when it is &quot;  −      5   and around it&quot;.

lilao8x · Accepted Answer

Note that       x    2    −  5  =      x    2    −            5        2    =  (  x  −      5    )  (  x  +      5    ), so we have  f  (  x  )  =                    x        2            −      5              x      +              5              =            (      x      −              5            )      (      x      +              5            )              x      +              5              =            x      −              5              1    =  x  −      5  However, this manipulation does not suddenly make   f defined when   x  =  −      5  . The only thing it does is that it simplifies the process of studying how   f behaves very close to   x  =  −      5  , namely, it behaves exactly like the function   g  (  x  )  =  x  −      5  , which is defined on the whole of the real line.

gvaldytist · Accepted Answer

The arithmetical expression                    x        2            −      5              x      +              5            is not defined for   x  =  −      5  To define a function, we must specify the domain. A definition like  f  (  x  )  =                    x        2            −      5              x      +              5            makes no sense because the domain is not specified. Think that   x could be a sheep, or a chair.On the other hand, a definition like  f  (  x  )  =                    x        2            −      5              x      +              5               for   x  ∈      R  doesn&#039;t either make sense, because the expression has no sense for an element of the domain (namely,   −      5  ).This definition is correct:  f  (  x  )  =                    x        2            −      5              x      +              5               for   x  ∈      R    ∖  {  −      5    }This definition is also correct, and defines the same function:  f  (  x  )  =  x  −      5     for   x  ∈      R    ∖  {  −      5    }Lastly, another correct definition, but this time it does not define the same function, because the domain is not the same:  g  (  x  )  =  x  −      5     for   x  ∈      R

Reduce a fraction Given the function f ( x ) = x

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