Questions: Define the domain of (s(x)=fracx+29(x-6)(x+7)). Use interval notation for your answer. The domain of (s(x)) is:

$Define the domain of (s(x)=fracx+29(x-6)(x+7)). Use interval notation for your answer. The domain of (s(x)) is:$

Transcript text: Define the domain of $s(x)=\frac{x+29}{(x-6)(x+7)}$. Use interval notation for your answer. The domain of $s(x)$ is: $\square$

Solution

Solution Steps

To find the domain of the function $ s(x) = \frac{x+29}{(x-6)(x+7)} $, we need to determine the values of $ x $ for which the function is defined. The function is undefined where the denominator is zero. Therefore, we need to find the values of $ x $ that make $ (x-6)(x+7) = 0 $ and exclude them from the domain.

Step 1: Identify the Function

We are given the function $ s(x) = \frac{x + 29}{(x - 6)(x + 7)} $. To determine the domain, we need to find the values of $ x $ that make the denominator zero.

Step 2: Find the Zeros of the Denominator

The denominator is given by $ (x - 6)(x + 7) $. Setting this equal to zero, we solve: \[ (x - 6)(x + 7) = 0 \] This gives us the solutions: \[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \] \[ x + 7 = 0 \quad \Rightarrow \quad x = -7 \]

Step 3: Determine the Domain

The function $ s(x) $ is undefined at $ x = 6 $ and $ x = -7 $. Therefore, we exclude these points from the domain. The domain in interval notation is: \[ (-\infty, -7) \cup (-7, 6) \cup (6, \infty) \]