f(x)=2x^2−x−6



g(x)=x^2+3x−10


What is (f/g)(x)
? Then, find the domain of (f/g)(x)
.

very short explanation, simple and sweet

1 answer

To find \((f/g)(x)\), we divide \(f(x)\) by \(g(x)\):

\[ (f/g)(x) = \frac{f(x)}{g(x)} = \frac{2x^2 - x - 6}{x^2 + 3x - 10} \]

Next, we need to determine the domain of \((f/g)(x)\). This is restricted by where \(g(x) \neq 0\).

First, set \(g(x) = 0\):

\[ x^2 + 3x - 10 = 0 \]

Factoring, we get:

\[ (x + 5)(x - 2) = 0 \]

Thus, \(g(x) = 0\) when \(x = -5\) or \(x = 2\).

The domain of \((f/g)(x)\) is all real numbers except \(x = -5\) and \(x = 2\):

\[ \text{Domain: } \mathbb{R} \setminus {-5, 2} \]