find the domain of the real valued function;

f(x) = sqrt(5 - (sqrtx))

my solution:

5 - (sqrt x) >=0
-(sqrt x) >= -5
(sqrt x) <= 5
x <= 25

(-infinity, 25]

is this correct?

no, since when x < 0, sqrt(x) is not defined. so, neither is sqrt(5-sqrt(x))

the domain is just [0,25]

thank you very much!

Yes, your solution is correct. The domain of the function f(x) = sqrt(5 - sqrt(x)) is (-infinity, 25]. The reason for this is because in order for the function to be defined, the expression inside the square root must be non-negative. Thus, we have 5 - sqrt(x) >= 0. Solving this inequality, we find sqrt(x) <= 5, which implies x <= 25. Therefore, the domain of the function is (-infinity, 25].

Yes, your solution and answer are correct. To find the domain of the function f(x) = sqrt(5 - sqrt(x)), you started by setting the expression inside the square root greater than or equal to 0:

5 - sqrt(x) ≥ 0

Then, you isolated the square root term by subtracting 5 from both sides:

-sqrt(x) ≥ -5

To get rid of the negative sign in front of the square root, you multiplied both sides of the inequality by -1, which reverses the inequality:

sqrt(x) ≤ 5

Next, you squared both sides of the inequality:

x ≤ 25

Lastly, you concluded that the domain of the function is all real numbers less than or equal to 25:

(-∞, 25]