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