### Instruction: Find the range of the function
\[m(x) = \sqrt{x + 5} + \sqrt{20 - x}.\].### Response: To find the range of a function, I need to figure out what values of y are possible outputs of the function for some value of x in the domain.
The domain of the function is the set of values of x that make the function well-defined and real-valued.
In this case, the function involves square roots, which are undefined for negative arguments and real for non-negative arguments.
Therefore, the domain of the function is the set of values of x that satisfy both x + 5 ≥ 0 and 20 - x ≥ 0.
These inequalities are equivalent to -5 ≤ x ≤ 20.
So the domain of the function is the closed interval [-5, 20].
To find the range, I can try to visualize the graph of the function or use some algebraic techniques.
I notice that the function is symmetric about x = 7.5, since m(7.5 + a) = m(7.5 - a) for any a in the domain.
This means that the function has a maximum value at x = 7.5, and it is decreasing on both sides of that point.
To find the maximum value, I can plug in x = 7.5 and simplify.
I get m(7.5) = √(7.5 + 5) + √(20 - 7.5) = √12.5 + √12.5 = 2√12.5 ≈ 7.07.
So the maximum possible output of the function is 2√12.5.
To find the minimum value, I need to consider the endpoints of the domain, where the function is either increasing or decreasing.
At x = -5, I get m(-5) = √(-5 + 5) + √(20 - (-5)) = 0 + √25 = 5.
At x = 20, I get m(20) = √(20 + 5) + √(20 - 20) = √25 + 0 = 5.
So the minimum possible output of the function is 5.
Therefore, the range of the function is the closed interval [5, 2√12.5].