15. If \( f^{\prime}(x)=\frac{(x+3)-(x+1)}{(x+3)^{2}} \) is a derivative of a function obtained by using the Quotient Rule, determine the original function \( f(x)-(2 \) points) 16. Let \( h(x) \) be a function defined for all \( x \neq 0 \) such that \( h(4)=-3 \) and the derivative of \( h \) is given by

To find the original function \( f(x) \) given the derivative \( f'(x) = \frac{(x+3)-(x+1)}{(x+3)^{2}} \), we should simplify this expression first. Simplifying gives us \( f'(x) = \frac{2}{(x+3)^{2}} \). To find \( f(x) \), we integrate \( f'(x) \): \[ f(x) = \int \frac{2}{(x+3)^{2}} \, dx = -\frac{2}{x+3} + C \] where \( C \) is the constant of integration. Thus, the original function is \( f(x) = -\frac{2}{x+3} + C \). Now regarding the function \( h(x) \), knowing \( h(4)=-3 \) provides a specific point that is often useful in determining the integration constant when we have the derivative, which can be integrated similarly to the previous function. The value of the function at a point helps to establish the constant \( C \) for \( h(x) \) once we find a general formula for \( h(x) \). Integrating \( h'(x) \) will yield a function format, allowing us to utilize the value of \( h(4) \) accurately.