Questions: Which graph represents the function of f(x)=(9 x^2+9 x-18)/(3 x+6)?

Which graph represents the function of f(x)=(9 x^2+9 x-18)/(3 x+6)?
Transcript text: Which graph represents the function of $f(x)=\frac{9 x^{2}+9 x-18}{3 x+6}$ ?
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Solution

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Solution Steps

Step 1: Simplify the Function

First, simplify the given function \( f(x) = \frac{9x^2 + 9x - 18}{3x + 6} \).

Factor the numerator and the denominator: \[ 9x^2 + 9x - 18 = 9(x^2 + x - 2) = 9(x + 2)(x - 1) \] \[ 3x + 6 = 3(x + 2) \]

So, the function simplifies to: \[ f(x) = \frac{9(x + 2)(x - 1)}{3(x + 2)} \]

Step 2: Cancel Common Factors

Cancel the common factor \((x + 2)\) in the numerator and the denominator: \[ f(x) = \frac{9(x - 1)}{3} = 3(x - 1) \]

Step 3: Identify the Simplified Function

The simplified function is: \[ f(x) = 3x - 3 \]

Step 4: Determine the Graph

The function \( f(x) = 3x - 3 \) is a linear function with a slope of 3 and a y-intercept of -3.

Step 5: Match the Graph

Compare the simplified function \( f(x) = 3x - 3 \) with the given graphs. The correct graph should have a slope of 3 and a y-intercept of -3.

Final Answer

The correct graph is the second one, which represents the function \( f(x) = 3x - 3 \).

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