Question

Which graph represents the function of f(x) = the quantity of 9 x squared minus 36, all over 3 x plus 6?

graph of 3 x plus 6, with discontinuity at 2, 12
graph of 3 x plus 6, with discontinuity at negative 2, 0
graph of 3 x minus 6, with discontinuity at negative 2, negative 12
graph of 3 x minus 6, with discontinuity at 2, 0

Answer 1

To analyze the function \( f(x) = \frac{9x^2 - 36}{3x + 6} \), we begin by simplifying the expression.

1. **Factoring the numerator**:
The numerator, \( 9x^2 - 36 \), is a difference of squares, which can be factored as follows:
\[
9x^2 - 36 = 9(x^2 - 4) = 9(x - 2)(x + 2)
\]

2. **Factoring the denominator**:
The denominator, \( 3x + 6 \), can be factored as:
\[
3x + 6 = 3(x + 2)
\]

3. **Simplifying the function**:
Thus, we can rewrite the function:
\[
f(x) = \frac{9(x - 2)(x + 2)}{3(x + 2)} = 3(x - 2)
\]
for \( x \neq -2 \), because at \( x = -2 \) the original function is undefined due to division by zero.

4. **Identifying the discontinuity**:
Since there is a factor \( x + 2 \) in both the numerator and denominator, the function is simplified to \( f(x) = 3(x - 2) \) for all \( x \) except \( x = -2 \). Therefore, there is a discontinuity at \( x = -2 \).

5. **Evaluating the value at the discontinuity**:
At the discontinuity \( x = -2 \), we can evaluate:
\[
f(-2) = 3(-2 - 2) = 3(-4) = -12
\]

Now that we have simplified the function and determined that there is a discontinuity at \( x = -2 \) with a corresponding value of \( y = -12 \), we can identify the correct graph.

The correct choice based on this analysis is:
- **Graph of \( 3x - 6 \), with discontinuity at \( -2, -12 \)**.