Questions: Refer to functions n and q. Evaluate (q * n)(x) and write the domain in interval notation. Write your answers as integers or simplified fractions. n(x) = x + 3, q(x) = 1 / (x + 8) Part 1 of 2 (q * n)(x) =

$Refer to functions n and q. Evaluate (q * n)(x) and write the domain in interval notation. Write your answers as integers or simplified fractions. n(x) = x + 3, q(x) = 1 / (x + 8) Part 1 of 2 (q * n)(x) =$

Transcript text: Refer to functions $n$ and $q$. Evaluate $(q \cdot n)(x)$ and write the domain in interval notation. Write your answers as integers or simplified fractions. \[ n(x)=x+3 \quad q(x)=\frac{1}{x+8} \] Part: $0 / 2$ Part 1 of 2 \[ (q \cdot n)(x)= \] $\square$

Solution

Solution Steps

To evaluate $(q \cdot n)(x)$, we need to find the product of the functions $n(x)$ and $q(x)$. This involves multiplying the expressions for $n(x)$ and $q(x)$. After finding the product, we will determine the domain by identifying any restrictions on $x$ that would make the expression undefined, such as division by zero.

Step 1: Evaluate the Product

We start by evaluating the product of the functions $n(x)$ and $q(x)$: \[ (q \cdot n)(x) = q(x) \cdot n(x) = \left(\frac{1}{x + 8}\right) \cdot (x + 3) = \frac{x + 3}{x + 8} \]

Step 2: Determine the Domain

Next, we need to find the domain of the function $(q \cdot n)(x)$. The function is undefined when the denominator is zero: \[ x + 8 = 0 \implies x = -8 \] Thus, the domain excludes $x = -8$. In interval notation, the domain is: \[ (-\infty, -8) \cup (-8, \infty) \]