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1)What is the domain of the function \( f(x) = \sqrt{3x+6} - 7\)? Answer in interval notation.

 

2)Express the domain of \(f(x) = \dfrac{\sqrt{-2x+7}}{x}\) in interval notation.

 

3)Express the domain of \(f(x) = \dfrac{\sqrt{x+1}}{(x-2)(x-4)}\) in interval notation.

 Sep 14, 2019
 #1
avatar+2856 
+2

1. the range of x is going to be like a ray, with a end point while goes to infinity in another direction.

 

\(f(x)=\sqrt{{\color{red}3}x+{\color{red}6}}-7\)

 

You do \((-1)*\frac{{\color{red}6}}{{\color{red}3}}\Rightarrow-2\)

 

This means the minimum of x is -2, while the maximum goes to infinity.

 Sep 14, 2019
 #2
avatar+111456 
+2

2.       √[-2x + 7]

          ________

                x

 

First of all....note that x CANNOT  =  0   because this would mean that we are dividing by 0 which is undefined

 

Also....under the radical.....    -2x + 7  must be   ≥ 0

 

So

 

-2x + 7  ≥  0

 

-2x  ≥ - 7       divide both sides by -2   and reverse the inequality sign

 

x  ≤ 7/2

 

So....the domain is

 

(-infinity, 0)  U  ( 0, 7/2)

 

See the graph here to confirm this :  https://www.desmos.com/calculator/qg90mexgvo

 

 

 

cool cool cool

 Sep 14, 2019
 #3
avatar+111456 
+2

3.      √ [ x + 1]

        ___________

          (x - 2) (x - 4)

 

First note that   if x = 2   or x = 4......the denominator  = 0....so.....these two values are not in the domain

 

Also...under the radical.....x + 1  must be  ≥  0

 

So

 

x + 1  ≥ 0

 

x  ≥ -1

 

So   the domain is

 

[ -1, 2)  U (2, 4)  U (4, infinity)

 

See the graph here : https://www.desmos.com/calculator/a4regqy27f

 

 

cool cool cool

 Sep 14, 2019

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