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.