Let's solve the two parts of the given problem.

Given function:

$f(x) = 6x - 2$

Part (a)

Find $f(x+h)$ and simplify.

To find $f(x+h)$, substitute $x+h$ into the function $f(x)$:

$f(x+h) = 6(x+h) - 2$ Now, expand and simplify:

$f(x+h) = 6x + 6h - 2$

Thus, the simplified expression for $f(x+h)$ is: $f(x+h) = 6x + 6h - 2$

Part (b)

Find $\frac{f(x+h) - f(x)}{h}$ and simplify.

We already have $f(x+h) = 6x + 6h - 2$ and $f(x) = 6x - 2$. Now, calculate $f(x+h) - f(x)$:

$f(x+h) - f(x) = (6x + 6h - 2) - (6x - 2)$

Simplifying: $f(x+h) - f(x) = 6x + 6h - 2 - 6x + 2 = 6h$

Now divide by $h$:

$\frac{f(x+h) - f(x)}{h} = \frac{6h}{h} = 6$

So the simplified expression for $\frac{f(x+h) - f(x)}{h}$ is: $6$

Would you like further explanation or any additional details?

Here are some related questions you could explore:

1. What is the interpretation of $\frac{f(x+h) - f(x)}{h}$ in calculus?
2. How does the expression $\frac{f(x+h) - f(x)}{h}$ relate to the derivative of a function?
3. What is the general form for finding $f(x+h)$ for any function $f(x)$?
4. How would the result change if $f(x)$ were a quadratic function?
5. What happens if $h$ approaches 0 in the expression $\frac{f(x+h) - f(x)}{h}$?

Tip: When simplifying expressions with $h$, remember that terms independent of $h$ often cancel out, making the process easier.