Is the function given by \( \mathrm{g}(\mathrm{x})=\frac{1}{\mathrm{x}+7} \) continuous over the interval \( (-5,5) \) ?

To determine if the function \( g(x) = \frac{1}{x + 7} \) is continuous over the interval \( (-5, 5) \), we need to check if there are any points in this interval where the function is undefined. The function \( g(x) \) is a rational function, and it is continuous wherever its denominator is not zero. The denominator of \( g(x) \) is \( x + 7 \). We need to find when this denominator is equal to zero: \[ x + 7 = 0 \implies x = -7 \] Now, we check if \( -7 \) is within the interval \( (-5, 5) \). The interval \( (-5, 5) \) includes all numbers greater than \(-5\) and less than \(5\). Since \(-7\) is less than \(-5\), it is not in the interval \( (-5, 5) \). Since the function \( g(x) \) is defined for all \( x \) in the interval \( (-5, 5) \) (as there are no points where the denominator is zero in this interval), we conclude that \( g(x) \) is continuous over \( (-5, 5) \). Thus, the answer is: \[ \text{Yes, } g(x) \text{ is continuous over the interval } (-5, 5). \]