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What is the domain of the real-valued function $f(x)=\frac{2x-7}{\sqrt{x^2-5x+6}}$?

Guest Jul 13, 2018
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\(f(x)=\frac{2x-7}{\sqrt{x^2-5x+6}} \)

 

We only need to worry about the denominator, since all real numbers will "work" for the numerator

 

Note that  the function inside the square root in the denominator must be  >  0  

 

So...let's find this

 

x^2 - 5x  + 6  >  0     factor

 

(x - 3) ( x - 2)  >  0

 

Setting each factor to 0  and solving for x produces  x  = 2   and  x  = 3

 

So....we have   three  possible  intervals  for solutions

 

(-inf, 2)   and   [2, 3]   and (3, infinity)

 

Note that  any  value of x  in the middle interval will make the inequality  false

 

So.....the domain  is       x < 2  ∪  x > 3

 

Here's a graph that will confirm this :

 

https://www.desmos.com/calculator/f0cw1ybcym

 

 

cool cool cool

CPhill  Jul 13, 2018

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