5. \( y=\frac{x-1}{x^{2}-16} \) (a) \( [x \in \mathbf{R} \mid x \neq 4\} \) () \( [x \in \mathbf{R} \mid x \neq 1] \) ( \( [x \in \mathbf{R} \mid x \neq 0] \) () \( [x \in \mathbf{R} \mid x \neq 4] \)

The function \( y = \frac{x-1}{x^2-16} \) has some interesting features. The denominator \( x^2 - 16 \) factors into \( (x - 4)(x + 4) \). This means that the function is undefined at \( x = 4 \) and \( x = -4 \). So, while you could say \( y \) is defined everywhere except at those points, the cleaner notation for the domain would be to specify what values of \( x \) are permissible! In terms of real-world application, understanding functions like this can help in various fields. For instance, if you were modeling a scenario where the value of \( y \) changes based on a certain input \( x \) — perhaps representing a system's efficiency or resource usage — knowing where the function is undefined can prevent errors in calculations and lead to better design or analysis decisions in engineering or economics!