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

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

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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

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CPhill · Answer 1 · 2018-03-19T05:03:39

[ 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)

CPhill · Answer 2 · 2018-03-19T05:44:15

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)

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