Question

Which graph represents the function of f(x) = the quantity of 9 x squared plus 9 x minus 18, 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 3, with discontinuity at negative 2, negative 9
graph of 3 x minus 3, with discontinuity at 2, 3

Answer 1

To determine which graph represents the function \( f(x) = \frac{9x^2 + 9x - 18}{3x + 6} \), we can start by simplifying the expression.

1. **Factor the numerator**:
The expression in the numerator is \( 9x^2 + 9x - 18 \). We can factor it:
\[
9x^2 + 9x - 18 = 9(x^2 + x - 2)
\]
Now, we need to factor \( x^2 + x - 2 \):
\[
x^2 + x - 2 = (x - 1)(x + 2)
\]
Therefore,
\[
9x^2 + 9x - 18 = 9(x - 1)(x + 2)
\]

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

3. **Simplify the function**:
Now substituting these factors back into the function:
\[
f(x) = \frac{9(x - 1)(x + 2)}{3(x + 2)}
\]
We see that \( (x + 2) \) cancels out (provided \( x \neq -2 \)):
\[
f(x) = 3(x - 1) \quad \text{for } x \neq -2
\]
Thus, the function is linear except at \( x = -2 \), where there is a discontinuity.

4. **Analyze the function**:
The function \( f(x) = 3(x - 1) \) is a straight line with a slope of 3.
- There will be a discontinuity at \( x = -2 \), where the function is undefined.
- The function will have a y-intercept when evaluated:
- \( f(0) = 3(0 - 1) = -3 \).

Based on the analysis:

- The discontinuity is at \( x = -2 \).
- The function's behavior reflects a linear graph, which is \( 3(x - 1) = 3x - 3 \).

Now, matching this information with the options provided, we look for:

- A graph of \( 3x - 3 \) with a discontinuity at \( x = -2 \).

This corresponds to:
- **Graph of 3x minus 3, with discontinuity at negative 2, negative 9**.

Thus, the correct answer is:
**The graph of \( 3x - 3 \), with discontinuity at \( -2, -9\)**.