Questions: Find the domain of the following rational function. R(x) = 12x / (x+9)

Transcript text: Find the domain of the following rational function. \[ R(x)=\frac{12 x}{x+9} \]

Solution

Solution Steps

To find the domain of the rational function \( R(x) = \frac{12x}{x+9} \), we need to determine the values of \( x \) for which the function is defined. The function is undefined when the denominator is zero. Therefore, we need to find the values of \( x \) that make the denominator zero and exclude them from the domain.

Step 1: Identify the Function

We are given the rational function \[ R(x) = \frac{12x}{x + 9}. \]

Step 2: Determine the Denominator

The function is undefined when the denominator is zero. Thus, we set the denominator equal to zero: \[ x + 9 = 0. \]

Step 3: Solve for Undefined Points

Solving the equation gives: \[ x = -9. \] This means the function is undefined at \( x = -9 \).

Step 4: Define the Domain

The domain of the function consists of all real numbers except the point where the function is undefined. Therefore, the domain can be expressed as: \[ \text{Domain} = (-\infty, -9) \cup (-9, \infty). \]

Final Answer

The domain of the function \( R(x) \) is \[ \boxed{(-\infty, -9) \cup (-9, \infty)}. \]

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