### 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 think about what possible values the output can take as the input varies.
I notice that this function involves square roots, so I have to be careful about the domain.
The expressions inside the square roots have to be nonnegative, or else the function is undefined.
That means I have two inequalities to satisfy: \(x + 5 \geq 0\) and \(20 - x \geq 0\).
Solving for x, I get \(-5 \leq x \leq 20\).
This is the domain of the function.
Now I want to see how the function behaves on this domain.
I can try plugging in some values of x and see what happens.
When x is -5, the first square root is 0 and the second one is 5.
So m(-5) = 0 + 5 = 5.
When x is 20, the first square root is 5 and the second one is 0.
So m(20) = 5 + 0 = 5.
That's interesting, the function has the same value at the endpoints of the domain.
I wonder if this is the minimum or the maximum value of the function, or both.
To find out, I can use some calculus and look for the critical points of the function.
A critical point is where the derivative is either zero or undefined.
To find the derivative, I can use the chain rule and the fact that \((\sqrt{u})' = \frac{u'}{2\sqrt{u}}\).
So the derivative of the function is
\[m'(x) = \frac{1}{2\sqrt{x + 5}} - \frac{1}{2\sqrt{20 - x}}.\]
This is defined for all x in the domain, so I don't have to worry about undefined points.
To find where the derivative is zero, I can set it equal to zero and solve for x.
\[m'(x) = 0 \implies \frac{1}{2\sqrt{x + 5}} - \frac{1}{2\sqrt{20 - x}} = 0.\]
I can multiply both sides by the common denominator \(2\sqrt{x + 5}\sqrt{20 - x}\) to get rid of the fractions.
\[\sqrt{x + 5} - \sqrt{20 - x} = 0.\]
I can isolate one of the square roots and square both sides to get rid of the radicals.
\[\sqrt{x + 5} = \sqrt{20 - x} \implies x + 5 = 20 - x \implies x = \frac{15}{2}.\]
This is the only critical point of the function, and it is in the domain.
I can plug it back into the function to find the corresponding output value.
\[m\left(\frac{15}{2}\right) = \sqrt{\frac{15}{2} + 5} + \sqrt{20 - \frac{15}{2}} = \sqrt{\frac{25}{2}} + \sqrt{\frac{25}{2}} = 2\sqrt{\frac{25}{2}}.\]
This is about 7.07, which is greater than 5.
So this is the maximum value of the function, and 5 is the minimum value.
Therefore, the range of the function is the interval \([5, 2\sqrt{25/2}]\).