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?

To determine which statement about the function \( f(x) = \frac{x - 7}{x^2 - 3x - 28} \) is true, we should analyze the function in terms of its domain, zeros, and any possible asymptotes or discontinuities.

### Step 1: Factor the Denominator
We need to factor the quadratic expression in the denominator:

\[
x^2 - 3x - 28
\]

To factor this, we look for two numbers that multiply to \(-28\) and add to \(-3\). These numbers are \(-7\) and \(4\). Therefore:

\[
x^2 - 3x - 28 = (x - 7)(x + 4)
\]

So the function can be rewritten as:

\[
f(x) = \frac{x - 7}{(x - 7)(x + 4)}
\]

### Step 2: Simplify the Function
Notice that the numerator \( x - 7 \) is the same as one of the factors in the denominator. This allows us to simplify the function, but we need to be careful about the domain:

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

### Step 3: Analyze the Function
- **Domain**: The function is undefined where the denominator is zero. Therefore, the function is undefined at \( x = 7 \) and \( x = -4 \).

- **Zeros**: The original function \( f(x) \) has no zeros because the simplified form \( \frac{1}{x+4} \) does not equal zero for any \( x \). 

- **Asymptotes**: 
  - **Vertical asymptote**: There is a vertical asymptote at \( x = -4 \) because as \( x \) approaches \(-4\), the function \( \frac{1}{x + 4} \) approaches infinity.
  - **Hole**: The function has a hole at \( x = 7 \), because the factor \( x - 7 \) was canceled during simplification, but \( f(x) \) is still undefined at \( x = 7 \).

### Conclusion
The most critical points are:
1. The function is undefined at \( x = -4 \) (vertical asymptote) and \( x = 7 \) (hole).
2. The simplified function \( \frac{1}{x + 4} \) suggests no zeros.

Now, based on typical choices, here is what might be true:
- The function has a vertical asymptote at \( x = -4 \).
- The function is undefined at \( x = 7 \) (a hole).
- The function does not have a zero.

If you have the actual answer choices, I can confirm the correct statement.

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Would you like any further details or clarification? Here are some related questions you might consider:

1. How do you identify vertical asymptotes in rational functions?
2. What are the steps to factor quadratic expressions?
3. Why does a hole occur in the graph of a function?
4. How does canceling factors in a rational expression affect the domain?
5. What are the types of discontinuities in a function?
6. How do you determine if a function has zeros?
7. What are the effects of asymptotes on the graph of a function?
8. How can you graph \( f(x) = \frac{1}{x + 4} \)?

**Tip**: Always check the domain of a function before and after simplifying it, as this helps to identify points of discontinuity like holes or vertical asymptotes.

Understand the function f(x) = (x - 7) / (x^2 - 3x - 28) with detailed steps on factoring the quadratic denominator, analyzing domain, vertical asymptotes, and discontinuities. Suitable for high school students and anyone studying rational functions.

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