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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.

Find the difference quotient of f; that is find (f(x + h) - f(x)) / h, h ≠ 0, for the function f(x) = √(x - 12). Hint: Rationalize the numerator.

This detailed guide explains how to find the difference quotient for the function f(x) = √(x - 12), including steps to rationalize the numerator and simplify the expression. Suitable for advanced high school mathematics students.

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