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3 questions

https://web2.0calc.com/questions/we-define-a-function-f-x-such-that-f-11-34-and-if

and

Find the domain of \(\frac{x^2 + 10x + 21}{x^2 + 4x - 21}\) (Express your answer using interval notation.)

and

Let \(f(x) = \left\lceil\dfrac{1}{x+2}\right\rceil\) for \(x > -2\), and \(f(x) = \left\lfloor\dfrac{1}{x+2}\right\rfloor\) for \(x < -2\). ( is not defined at \(x = -2\).) Which integer is not in the range of \(f(x)\)?

Guest Mar 18, 2018
edited by Guest  Mar 18, 2018
 #1
avatar+86861 
+3

[ x^2 + 10x  + 21]

______________

[ x^2 + 4x  -  21 ]  

 

 

The domain excludes any x values that makes the denominator  = 0

 

So

 

x^2 + 4x  -  21  =  0

(x + 7) ( x - 3)  = 0

 

Setting each factor to 0 and solving for x  produces the two x values not in the domain  ⇒

 

x  =  -  7          and  x   = 3

 

So.....the domain  is

 

(-inf, -7) U (-7, 3)  U (3, inf)

 

 

 

cool cool cool

CPhill  Mar 19, 2018
 #2
avatar+86861 
+3

f(x)   =  ceiling function  [ 1 / (x + 2) ]    x >-2

f(x)  =  floor function [ 1 / (x + 2) ]  x < -2

 

Notice that the first function can never  evaluate to 0  when x > -2

And the second function can never evaluate to 0  when x < - 2

 

So  0  is not in the range of f(x)

 

 

cool cool cool

CPhill  Mar 19, 2018

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