determine the domain of the real valued function f(x) = sqrt(5 - (sqrtx))
sqrt(x) has domain x >= 0
sqrt(5-sqrt(x)) has domain 5 - sqrt(x) >= 0, or sqrt(x) <= 5
so, the domain of f(x) is 0 <= x <= 25
To determine the domain of the function f(x) = sqrt(5 - (sqrtx)), we need to identify the values of x for which the function is defined.
First, notice that the function involves the square root of both 5 and (sqrtx). In order for the square root to be defined, the expression inside the square root, in this case 5 - (sqrtx), must be greater than or equal to zero.
So, we set up the inequality:
5 - (sqrtx) ≥ 0
To solve this inequality, we isolate the square root:
(sqrtx) ≤ 5
Next, we square both sides of the inequality to get rid of the square root:
x ≤ 25
However, this is not the entire domain of the function. There is still another condition to consider. The expression inside the square root, 5 - (sqrtx), must also be non-negative. Therefore, we set up another inequality:
5 - (sqrtx) ≥ 0
Solving this inequality, we isolate the square root:
(sqrtx) ≤ 5
Then we square both sides of the inequality:
x ≤ 25
Now, we need to find the intersection of the two inequalities we derived, x ≤ 25 and x ≤ 25. Since the value of x must satisfy both conditions, the domain of the function f(x) = sqrt(5 - (sqrtx)) is:
x ≤ 25
In other words, the domain consists of all real numbers x such that x is less than or equal to 25.