To find the expression \(\frac{f(x) - f(1)}{x - 1}\), we first need to calculate the function \(f(x)\) at \(x = 1\) and then substitute the values into the expression.

### Step 1: Evaluate \(f(1)\)
Given the function:

\[
f(x) = \frac{x + 7}{x + 3}
\]

Let's compute \(f(1)\):

\[
f(1) = \frac{1 + 7}{1 + 3} = \frac{8}{4} = 2
\]

### Step 2: Substitute into the Expression
Now, the expression we need to evaluate is:

\[
\frac{f(x) - f(1)}{x - 1} = \frac{\frac{x + 7}{x + 3} - 2}{x - 1}
\]

### Step 3: Simplify the Expression
Let's simplify the numerator:

\[
f(x) - f(1) = \frac{x + 7}{x + 3} - 2 = \frac{x + 7 - 2(x + 3)}{x + 3}
\]

Simplify further:

\[
f(x) - f(1) = \frac{x + 7 - 2x - 6}{x + 3} = \frac{-x + 1}{x + 3} = \frac{-(x - 1)}{x + 3}
\]

Thus, our expression becomes:

\[
\frac{f(x) - f(1)}{x - 1} = \frac{\frac{-(x - 1)}{x + 3}}{x - 1}
\]

### Step 4: Further Simplify
We can simplify this as:

\[
= \frac{-(x - 1)}{(x + 3)(x - 1)}
\]

Since \(x \neq 1\), we can cancel out the \((x - 1)\):

\[
= \frac{-1}{x + 3}
\]

### Final Answer
The simplified form of the expression is:

\[
\frac{f(x) - f(1)}{x - 1} = -\frac{1}{x + 3}
\]

Would you like more details, or do you have any questions?

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Here are 5 related questions for further exploration:

1. How do you find the derivative of a rational function using the limit definition?
2. What happens to the expression as \(x \to -3\)?
3. How can you find horizontal and vertical asymptotes for the function \(f(x) = \frac{x + 7}{x + 3}\)?
4. What are the critical points of \(f(x) = \frac{x + 7}{x + 3}\)?
5. How do you perform polynomial long division to simplify rational expressions?

**Tip:** Always check for values where the function is undefined, especially with rational expressions!

Learn how to evaluate the difference quotient of the rational function f(x) = (x + 7) / (x + 3). This problem involves computing the function value at a specific point and simplifying the resulting expression. Suitable for advanced high school students.

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