For the given function, find (f(x + h) -f(x)/h).

f(x ) = 6x - 6

The expression \( \frac{f(x + h) - f(x)}{h} \) is a common difference quotient used in calculus to compute the derivative of a function.

Given \( f(x) = 6x - 6 \), let's compute the difference quotient step by step:

### Step 1: Calculate \( f(x + h) \)
Substitute \( x + h \) into the function:
\[
f(x + h) = 6(x + h) - 6 = 6x + 6h - 6
\]

### Step 2: Compute \( f(x + h) - f(x) \)
Now subtract \( f(x) = 6x - 6 \) from \( f(x + h) \):
\[
f(x + h) - f(x) = (6x + 6h - 6) - (6x - 6) = 6h
\]

### Step 3: Divide by \( h \)
Now divide the result by \( h \):
\[
\frac{f(x + h) - f(x)}{h} = \frac{6h}{h} = 6
\]

Thus, the difference quotient \( \frac{f(x + h) - f(x)}{h} \) simplifies to **6**.

Would you like more details or have any questions?

### Related questions:
1. What is the significance of the difference quotient?
2. How is the difference quotient related to the derivative of a function?
3. How would this process change if \( f(x) \) were a quadratic function?
4. What happens to the difference quotient as \( h \to 0 \)?
5. How can the difference quotient help in understanding linear functions?

### Tip:
For linear functions like \( f(x) = 6x - 6 \), the difference quotient always simplifies to the constant slope of the line.

For the given function, find (f(x + h) - f(x))/h. f(x) = 6x - 6

Explore the solution to the difference quotient (f(x + h) - f(x)) / h for the linear function f(x) = 6x - 6. Understand the steps involved in simplifying this calculus problem, suitable for high school calculus students.

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