The problem asks for the difference quotient of the function $f(x) = \sqrt{x - 12}$, and gives the formula:

$\frac{f(x+h) - f(x)}{h}, \quad h \neq 0.$

Let's solve this step by step:

Step 1: Write the difference quotient

We need to substitute the function $f(x) = \sqrt{x - 12}$ into the difference quotient formula:

$\frac{f(x+h) - f(x)}{h} = \frac{\sqrt{(x+h) - 12} - \sqrt{x - 12}}{h}.$

This simplifies to:

$\frac{\sqrt{x + h - 12} - \sqrt{x - 12}}{h}.$

Step 2: Rationalize the numerator

To simplify this expression, we multiply the numerator and denominator by the conjugate of the numerator:

$\text{Multiply by } \frac{\sqrt{x + h - 12} + \sqrt{x - 12}}{\sqrt{x + h - 12} + \sqrt{x - 12}}.$

This gives:

$\frac{\left( \sqrt{x + h - 12} - \sqrt{x - 12} \right) \left( \sqrt{x + h - 12} + \sqrt{x - 12} \right)}{h \left( \sqrt{x + h - 12} + \sqrt{x - 12} \right)}.$

Using the difference of squares formula:

$(a - b)(a + b) = a^2 - b^2,$

we get:

$\frac{(x + h - 12) - (x - 12)}{h \left( \sqrt{x + h - 12} + \sqrt{x - 12} \right)}.$

Step 3: Simplify the numerator

Simplifying the expression inside the numerator:

$(x + h - 12) - (x - 12) = x + h - 12 - x + 12 = h.$

So the expression becomes:

$\frac{h}{h \left( \sqrt{x + h - 12} + \sqrt{x - 12} \right)}.$

Step 4: Cancel $h$

Now we can cancel $h$ from the numerator and denominator:

$\frac{1}{\sqrt{x + h - 12} + \sqrt{x - 12}}.$

Final Answer:

The simplified form of the difference quotient is:

$\frac{1}{\sqrt{x + h - 12} + \sqrt{x - 12}}.$

Would you like more details or have any questions?

Here are some related questions:

1. What is the purpose of rationalizing the numerator in the difference quotient?
2. How does the difference quotient relate to the derivative?
3. What happens if you don’t cancel out $h$ properly?
4. Can the difference quotient be applied to functions other than square roots?
5. How does this formula change if the function is $f(x) = \sqrt{x + 12}$?

Tip: When simplifying a difference quotient involving square roots, rationalizing the numerator is key to eliminating square roots and making further simplifications easier.