***The complete question is: Let f(x)= sqrt 6x and g(x)=x+3 smallest number that is in the domain of f(g(x))***
Answer: The smallest number that is in the domain is -3.
Explanation:
Given functions:
[tex]f(x) = \sqrt{6x}[/tex]
[tex]g(x) = x + 3[/tex]
To find f(g(x)), put g(x) in f(x) as follows:
[tex]f(g(x)) = f(x+3) = \sqrt{6(x+3)} = \sqrt{6x + 18}[/tex]
As (6x+18) is in square-root (and square-root of a negative number is an imaginary value), therefore, the domain should be the following:
[tex]6x + 18 \geq 0\\6x \geq -18\\x \geq -3[/tex]
Therefore, the domain is [-3, +∞).
So the smallest number that is in the domain is -3.
