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# help

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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 $$f(x)$$?

Mar 18, 2018
edited by Guest  Mar 18, 2018

#1
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[ 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)

Mar 19, 2018
#2
+100439
+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)

Mar 19, 2018