Questions: For this problem, there is no partial credit. Find the domain of the following function. Write your answer using interval notation. f(x) = (ln(15-x))/(x+2)

Solution Steps

To find the domain of the function \( f(x) = \frac{\ln(15-x)}{x+2} \), we need to determine the values of \( x \) for which the function is defined. The natural logarithm function \( \ln(15-x) \) is defined when \( 15-x > 0 \), and the denominator \( x+2 \) must not be zero.

1. Solve \( 15-x > 0 \) to find the range of \( x \) for which the logarithm is defined.
2. Solve \( x+2 \neq 0 \) to find the values of \( x \) that do not make the denominator zero.
3. Combine these conditions to find the domain of the function.

The function \( f(x) = \frac{\ln(15-x)}{x+2} \) requires that the argument of the logarithm is positive. Thus, we need to solve the inequality: \[ 15 - x > 0 \] This simplifies to: \[ x < 15 \]

Next, we need to ensure that the denominator does not equal zero: \[ x + 2 \neq 0 \] This simplifies to: \[ x \neq -2 \]

From the two conditions derived, we have:

1. \( x < 15 \)
2. \( x \neq -2 \)

The domain of the function is all real numbers less than 15, excluding -2. In interval notation, this can be expressed as: \[ (-\infty, -2) \cup (-2, 15) \]

Final Answer