Question

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

Answer 1

To analyze the function \( f(x) = \frac{x - 7}{x^2 - 3x - 28} \), we need to identify where the function is positive or negative.

First, let's factor the denominator:

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

To factor this quadratic, we look for two numbers that multiply to \(-28\) and add up to \(-3\). The correct factorization is:

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

Thus, we can rewrite the function as:

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

We can simplify this, noting that \( x \neq 7 \):

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

Next, we determine the sign of \( f(x) \) for various intervals:

1. **For \( x < -4 \)**:
- \( x + 4 < 0 \) ⇒ \( f(x) < 0 \)

2. **For \( -4 < x < 7 \)**:
- \( x + 4 > 0 \) ⇒ \( f(x) > 0 \)

3. **At \( x = 7 \)**:
- \( f(x) \) is undefined.

4. **For \( x > 7 \)**:
- \( x + 4 > 0 \) ⇒ \( f(x) > 0 \)

From this analysis, we can conclude:
- \( f(x) \) is negative for \( x < -4 \) and positive for \( -4 < x < 7 \).
- \( f(x) \) is positive for \( x > 7 \).

Now, let's evaluate the options presented:

1. **\( f(x) \) is positive for all \( x < 7 \):** This is false because \( f(x) < 0 \) for \( x < -4 \).
2. **\( f(x) \) is negative for all \( x < 7 \):** This is false because \( f(x) > 0 \) for \( -4 < x < 7 \).
3. **\( f(x) \) is positive for all \( x > –4 \):** This is false because \( f(x) < 0 \) for \( x < -4 \).
4. **\( f(x) \) is negative for all \( x > –4 \):** This is false because \( f(x) > 0 \) for \( -4 < x < 7 \) and also for \( x > 7 \).

Thus, none of the provided statements are true based on our analysis of \( f(x) \).