Find f(a + h) - f(a)
h
 and simplify where,
 (i) f(x) = 6x - 9 

We are tasked with finding the expression \(\frac{f(a+h) - f(a)}{h}\) and simplifying it for the given function \(f(x) = 6x - 9\).

### Step 1: Determine \(f(a + h)\)

Since \(f(x) = 6x - 9\), replace \(x\) with \(a + h\) to get:
\[
f(a + h) = 6(a + h) - 9 = 6a + 6h - 9
\]

### Step 2: Determine \(f(a)\)

Now, find \(f(a)\) by replacing \(x\) with \(a\) in the function \(f(x)\):
\[
f(a) = 6a - 9
\]

### Step 3: Calculate \(f(a + h) - f(a)\)

Now subtract \(f(a)\) from \(f(a + h)\):
\[
f(a + h) - f(a) = (6a + 6h - 9) - (6a - 9)
\]
Simplifying the expression:
\[
f(a + h) - f(a) = 6a + 6h - 9 - 6a + 9 = 6h
\]

### Step 4: Divide by \(h\)

Now, divide the result by \(h\):
\[
\frac{f(a + h) - f(a)}{h} = \frac{6h}{h} = 6
\]

### Final Answer:
\[
\frac{f(a+h) - f(a)}{h} = 6
\]

Would you like further details or have any questions?

Here are 5 related questions to expand on this:
1. How would the result change if \(f(x)\) were a quadratic function?
2. What is the significance of this expression in calculus?
3. Can you apply the same steps if \(f(x) = 3x^2 - 2x + 1\)?
4. How would you interpret this result geometrically?
5. What happens to the expression if \(h\) approaches 0?

**Tip:** The expression \(\frac{f(a+h) - f(a)}{h}\) is the difference quotient, a fundamental concept in finding the derivative of a function.

Find \(\frac{f(a + h) - f(a)}{h}\) and simplify where, (i) f(x) = 6x - 9

Explore how to simplify the difference quotient \(\frac{f(a + h) - f(a)}{h}\) for the linear function f(x) = 6x - 9. Learn step-by-step how this basic calculus concept works. Suitable for high school students in Grades 9-12.

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