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Compute the domain of the function $$f(x)=\frac{1}{\lfloor x^2-7x+13\rfloor}.$$

 Dec 16, 2020
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This one is a little hard to  think  about....

 

The function x^2 - 7x + 13   will  never   =  0   since it lies wholly above the  x axis

 

Since we  cant  divide  by  0, we actually  want to find the  x values  where 

x^2 -7x + 13  =  1

The two x values that  make this true will produce  a floor value of  1

The  two solutions that we  find will be  included  in the  domain....all real numbers  between these WILL NOT be in the  domain   since they  will make the floor value    = 0    and we  can't  divide  by  that

So  rearranging, we have 

 

x^2 -7x + 12   = 0     factor

 

(x - 3) ( x - 4)   =  0

 

Setting each factor to   0  and  solving for x produces  x  =3    and  x  = 4

 

So....the domain will be  (-inf. 3 ] U  [4, inf )

 

This graph seems to  confirm  our answer  : https://www.desmos.com/calculator/df9v0bh5wt

 

 

cool cool cool

 Dec 16, 2020
edited by CPhill  Dec 16, 2020

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