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The question is asking to construct and simplify the **difference quotient** for the function \( f(x) = x^2 - 2 \). The difference quotient is given by:

\[
\frac{f(x+h) - f(x)}{h}
\]

### Step-by-step solution:

1. **Substitute \( f(x) = x^2 - 2 \) into the difference quotient**:

\[
f(x+h) = (x+h)^2 - 2
\]
Simplify \( f(x+h) \):
\[
(x+h)^2 - 2 = x^2 + 2xh + h^2 - 2
\]

2. **Now, compute \( f(x+h) - f(x) \)**:
\[
f(x+h) - f(x) = (x^2 + 2xh + h^2 - 2) - (x^2 - 2)
\]
Simplify:
\[
f(x+h) - f(x) = x^2 + 2xh + h^2 - 2 - x^2 + 2 = 2xh + h^2
\]

3. **Divide by \( h \)**:
\[
\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2}{h}
\]
Simplify:
\[
\frac{f(x+h) - f(x)}{h} = 2x + h
\]

### Final Answer:
The simplified difference quotient is:

\[
2x + h
\]

Would you like further details or explanations on any part of this process?

Here are some related questions for practice:
1. What is the difference quotient for \( f(x) = 3x^2 - 4x + 1 \)?
2. How does the difference quotient relate to the derivative of a function?
3. What happens to the difference quotient as \( h \) approaches 0?
4. Can the difference quotient be used to find the slope of a secant line? Why?
5. What are some other applications of the difference quotient in calculus?

**Tip:** The difference quotient helps estimate the slope of the tangent line to a curve at any point and is crucial in finding derivatives.

For the function f(x) = x^2 - 2, construct and simplify the difference quotient: [f(x+h) - f(x)] / h.

Learn how to construct and simplify the difference quotient for the quadratic function f(x) = x^2 - 2. This solution explains step-by-step how to apply the difference quotient formula, suitable for students in Grades 10-12.

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