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.

Suspect Sep 14, 2019

#1**+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.

CalculatorUser Sep 14, 2019

#2**+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

CPhill Sep 14, 2019

#3**+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

CPhill Sep 14, 2019