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1. Let \[f(x) = \left\{ \begin{array}{cl} 2x + 1 & \text{if } x \le 3, \\ 8 - 4x & \text{if } x > 3. \end{array} \right.\]Find the sum of all values of $x$ such that $f(x) = 0.$

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

 Aug 21, 2018
 #1
avatar+98130 
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\(f(x) = \left\{ \begin{array}{cl} 2x + 1 & \text{if } x \le 3, \\ 8 - 4x & \text{if } x > 3. \end{array} \right.\ \)

 

Find the sum of all values of x such that f(x) = 0

 

0  =  2x +  1    subtract  1  from each side

-1  = 2x           divide both sides by 2

-1/2  = x

 

And

0  = 8 - 4x     add 4x to both sides

4x  = 8          divede both sides   by  4

x  =  2       [  not  in the domain  of the second function ]

 

So  the  sum  of all  values  such that f(x)  = 0   is  just   -1/2

 

 

cool cool  cool

 Aug 21, 2018
 #2
avatar+98130 
+1

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

 

The domain will only be restricted by all the values of x that make the  denominator  = 0 

 

So we have

 

x^2 + 4x  - 21    =  0       factor

 

(x + 7)  ( x - 3)   =   0

 

Set  both  factors to 0  and solve for x   and we havethat   x  = -7   and  x = 3

 

So....the domain is   (-infinity, -7) U ( -7, 3) U (3, infinity)

 

 

cool cool cool

 Aug 21, 2018

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