The given problem involves finding the rate of change of a function \( f(x) \), with two conditions:

1. \( f(1) = -1 \)
2. \( f(x) = f(x - 1) + 7 \)

### Step 1: Recognize the pattern
The recurrence relation \( f(x) = f(x-1) + 7 \) suggests that the function increases by 7 each time \( x \) increases by 1.

### Step 2: Find the general formula
Let's compute a few values of \( f(x) \) based on this rule:
- \( f(1) = -1 \)
- \( f(2) = f(1) + 7 = -1 + 7 = 6 \)
- \( f(3) = f(2) + 7 = 6 + 7 = 13 \)
- \( f(4) = f(3) + 7 = 13 + 7 = 20 \)

We can see a pattern that suggests \( f(x) \) follows the equation of a linear function.

### Step 3: General form of the function
The recurrence relation tells us that each increment of \( x \) increases \( f(x) \) by 7. Thus, we can express \( f(x) \) as a linear function of the form:

\[
f(x) = 7x + C
\]

### Step 4: Find the constant \( C \)
Using the condition \( f(1) = -1 \), substitute \( x = 1 \) into the equation:

\[
-1 = 7(1) + C
\]

Solving for \( C \):

\[
-1 = 7 + C \quad \Rightarrow \quad C = -8
\]

### Step 5: Final equation for \( f(x) \)
Thus, the general form of the function is:

\[
f(x) = 7x - 8
\]

### Step 6: Rate of change
The rate of change of a linear function \( f(x) = mx + b \) is given by the coefficient of \( x \), which is 7. Therefore, the rate of change of \( f(x) \) is:

\[
\text{Rate of change} = 7
\]

### Final Answer:
The rate of change of the function is 7.

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Would you like a more detailed explanation or further clarification?

Here are five related questions:
1. How do we calculate the rate of change for non-linear functions?
2. Can we use similar recurrence relations for quadratic functions?
3. How would the solution change if \( f(x) = f(x-1) + c \) where \( c \) is not constant?
4. What is the geometric interpretation of the rate of change in linear functions?
5. How does the initial condition \( f(1) = -1 \) influence the final function?

**Tip**: When working with recurrence relations, always try to find a general formula by recognizing patterns or applying known types of sequences (like arithmetic sequences).

Learn how to find the rate of change for the recurrence relation f(x) = f(x-1) + 7 with the initial condition f(1) = -1. The solution involves key concepts in algebra and linear functions, suitable for students in Grades 8-10.

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