Given f of x is equal to the quantity x minus 7 end quantity divided by the quantity x squared minus 3x minus 28 end quantity, which of the following is true?

 f(x) is positive for all x < 7
 f(x) is negative for all x < 7
 f(x) is positive for all x > –4
 f(x) is negative for all x > –4

Given the function \( f(x) = \frac{x - 7}{x^2 - 3x - 28} \), we already factored and simplified the expression as follows:

\[
f(x) = \frac{x - 7}{(x - 7)(x + 4)} = \frac{1}{x + 4}, \quad \text{for } x \neq 7
\]

Let's evaluate each statement to determine which one is true.

### Step 1: Consider the simplified form of \( f(x) \):
\[ f(x) = \frac{1}{x + 4} \]

### Step 2: Determine the sign of \( f(x) \) based on the value of \( x \):

- **For \( x < -4 \)**:
  - \( x + 4 \) is negative, so \( f(x) = \frac{1}{x + 4} \) is negative.

- **For \( x > -4 \)**:
  - \( x + 4 \) is positive, so \( f(x) = \frac{1}{x + 4} \) is positive.

### Step 3: Evaluate the Statements

1. **\( f(x) \) is positive for all \( x < 7 \):**
   - This is **false**. \( f(x) \) is negative for \( x < -4 \).

2. **\( f(x) \) is negative for all \( x < 7 \):**
   - This is **false**. \( f(x) \) is negative only for \( x < -4 \).

3. **\( f(x) \) is positive for all \( x > -4 \):**
   - This is **true**. For all \( x > -4 \), \( f(x) = \frac{1}{x + 4} \) is positive.

4. **\( f(x) \) is negative for all \( x > -4 \):**
   - This is **false**. \( f(x) \) is positive for \( x > -4 \).

### Conclusion:
The correct statement is **\( f(x) \) is positive for all \( x > -4 \)**.

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Would you like any more details or have any other questions? Here are some related questions you might consider:

1. How do you determine the intervals where a function is positive or negative?
2. What is the effect of a vertical asymptote on the sign of a function?
3. How does simplifying a rational function help in analyzing its properties?
4. What role do zeros and undefined points play in the behavior of a function?
5. How can the graph of \( \frac{1}{x + 4} \) be sketched and analyzed?
6. What happens to \( f(x) \) as \( x \) approaches \( -4 \) from both sides?
7. How do you find the domain and range of rational functions?
8. What is the significance of factoring the denominator in rational functions?

**Tip**: When analyzing the sign of a rational function, always consider the sign changes that occur at key points like zeros, vertical asymptotes, and points where the function is undefined.

Explore the analysis of the rational function f(x) = (x - 7) / (x^2 - 3x - 28) to determine when it is positive or negative. Learn about rational function simplification and the impact of different x values on the function's sign. Suitable for high school students studying functions and algebra.

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