Bot GPT 4-o mini To determine which graph represents the function \( f(x) = \frac{9x^2 + 9x - 18}{3x + 6} \), we can start by simplifying the expression.
-
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, \[ 9x^2 + 9x - 18 = 9(x - 1)(x + 2) \]
-
Factor the denominator: The denominator \( 3x + 6 \) can be factored as: \[ 3x + 6 = 3(x + 2) \]
-
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.
-
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 \( -2, -9\).
1 answer