Respuesta :
We want to find the domain of
[tex] f(x) = \sqrt{9-x} [/tex]
The domain of a function is the set of points (in this case, of numbers) which the function accepts as inputs, i.e. the set of numbers for which the function is capable of computing the output.
In this case, the function involves a square root, and square roots are not always computable using real number.
In fact, you can't find any real number to satisfy something like
[tex] \sqrt{-5} = x [/tex]
because it would imply
[tex] x^2 = -5 [/tex]
which is impossible using real numbers, because all real numbers become positive when squared.
So, since we can't compute the square root of a negative number, we must make sure that this doesn't happen. We're computing the square root of nine minus x, so that quantity must be positive:
[tex] 9-x > 0 \iff x < 9 [/tex]
So, if you choose any number smaller than 9, the quantity [tex]9-x[/tex] is guaranteed to be positive, and we are sure that we can compute its square root.
If you recall what we said at the beginning, thisi is exactly the definition of the domain!
So, the domain of the function is the set
[tex] \{ x \in \mathbb{R}\ :\ x<9 \}[/tex]
