Questions: State the domain of the function F(x) = (x+6)/(x^2 + 4x + 3) Interval notation

Transcript text: State The domain of the function \[ F(x)=\frac{x+6}{x^{2}+4 x+3} \] Interval notation

Solution

Solution Steps

Step 1: Identify the Denominator Polynomial \(D(x)\)

The denominator polynomial is \(D(x) = x^{2} + 4 x + 3\).

Step 2: Find the Roots of the Denominator Polynomial

Solve \(D(x) = 0\) to find the roots. The roots are \(x = \left\{-3, -1\right\}\).

Step 3: Exclude the Roots from the Real Number Set

The domain of \(F(x) = \frac{N(x)}{D(x)}\) consists of all real numbers except \(x = \left\{-3, -1\right\}\), which means the domain is \(\mathbb{R} - \{-3, -1\}\).

Step 4: Consider the Numerator Polynomial

While the numerator polynomial \(N(x) = x + 6\) does not affect the domain directly, it's important for understanding the behavior of the function.

Step 5: Check for Special Cases

If \(N(x)\) and \(D(x)\) have a common factor, it could potentially simplify the function. However, for the purpose of finding the domain, we focus on the roots of \(D(x)\) only.