\begin{tabular}{l} What is the domain of \( \left(\frac{f}{g}\right)(x) \) ? \\ \( \qquad \)\begin{tabular}{l}\( f(x)=-3 x+1 \) \\ \( g(x)=-5 x+3 \) \\ All real numbers except \( \frac{3}{5} \) \\ All real numbers except \( \frac{1}{3} \) \\ All real numbers except \( -\frac{1}{3} \) \\ All real numbers \\ \hline\end{tabular} \\ \hline\end{tabular}

To determine the domain of \( \left(\frac{f}{g}\right)(x) \), we need to ensure that \( g(x) \neq 0 \), because division by zero is undefined. First, let's solve for \( g(x) = 0 \): \[ -5x + 3 = 0 \quad \Rightarrow \quad -5x = -3 \quad \Rightarrow \quad x = \frac{3}{5} \] Thus, \( g(x) \) is zero when \( x = \frac{3}{5} \). Therefore, the domain of \( \left(\frac{f}{g}\right)(x) \) excludes \( x = \frac{3}{5} \) and includes all other real numbers. So the domain is all real numbers except \( \frac{3}{5} \). To further dive into function behavior, consider how the values of \( f(x) \) and \( g(x) \) interact near critical points. Since \( g(x) \) approaches zero at \( x = \frac{3}{5} \), close to this value, observe how \( \left(\frac{f}{g}\right)(x) \) can tend towards positive or negative infinity. Points of discontinuity often yield interesting characteristics for functions. Exploring rational functions in practical scenarios can lead to impressive insights, like in physics for distance-time graphs. Understanding the domain helps in determining when an equation might predict physical phenomena accurately. By knowing where functions behave unexpectedly, you can avoid misinterpretations in your calculations or models!