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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?

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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 \(\frac{f(x+h) - f(x)}{h}\)?

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

Given f(x) = 6x - 2, (a) Find f(x + h) and simplify. (b) Find (f(x + h) - f(x)) / h and simplify.

Learn how to simplify expressions like f(x+h) and the difference quotient (f(x+h) - f(x)) / h for the linear function f(x) = 6x - 2. This step-by-step guide covers core algebraic concepts, suitable for high school students in Grades 9-12.

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