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.
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.
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
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