Questions: Find the domain of the function. f(x)=6/(x+15)+7/(x-10) The domain of f(x) is (Type your answer in interval notation.)

Solution Steps

To find the domain of the function \( f(x) = \frac{6}{x+15} + \frac{7}{x-10} \), we need to determine the values of \( x \) for which the function is defined. The function is undefined where the denominators are zero. Therefore, we need to find the values of \( x \) that make \( x+15 = 0 \) and \( x-10 = 0 \).

1. Identify the values of \( x \) that make the denominators zero.
2. Exclude these values from the domain.
3. Express the domain in interval notation.

To find the domain of the function \( f(x) = \frac{6}{x+15} + \frac{7}{x-10} \), we first identify the values of \( x \) that make the denominators zero. Setting the denominators equal to zero gives us: \[ x + 15 = 0 \quad \Rightarrow \quad x = -15 \] \[ x - 10 = 0 \quad \Rightarrow \quad x = 10 \]

The function \( f(x) \) is undefined at \( x = -15 \) and \( x = 10 \). Therefore, these points must be excluded from the domain.

The domain of \( f(x) \) includes all real numbers except \( -15 \) and \( 10 \). In interval notation, this is expressed as: \[ (-\infty, -15) \cup (-15, 10) \cup (10, \infty) \]

Final Answer