To find the domain of the function \(\frac{f}{g}(x)\), where \(f(x) = x^2 - 3x - 10\) and \(g(x) = x^2 - 4\), we need to determine when \(g(x) \neq 0\) because a function is undefined when its denominator is zero.
First, let's solve for when \(g(x) = 0\):
\[ g(x) = x^2 - 4 = 0 \]
Factoring the quadratic, we find:
\[ (x - 2)(x + 2) = 0 \]
Setting each factor to zero gives us:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \] \[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
Thus, \(g(x) = 0\) when \(x = 2\) or \(x = -2\). Therefore, the function \(\frac{f}{g}(x)\) is undefined at these points.
The domain of the function \(\frac{f}{g}(x)\) is all real numbers except for these points. In interval notation, this is:
\[ \text{Domain} = { x ,|, x \neq -2 \text{ and } x \neq 2 } \]
Thus, the correct response is:
left-brace x vertical line x not equal to negative 2 or 2 right-brace
1 answer