6. \( y=\frac{x}{x^{2}-9 x} \) (a) \( \{x \in \mathbf{R} \mid x \neq 0, x=9\} \) (b) \( \{x \in \mathbf{R} \mid x \neq 0, x=-9\} \) (c) \( \{x \in \mathbf{R} \mid x \neq 0\} \) (d) \( [x \in \mathbf{R} \mid x \neq 9\} \)

To find the relevant information, we start by determining where the function \( y = \frac{x}{x^2 - 9x} \) is undefined. The denominator can be factored into \( x(x - 9) \). This means the function will be undefined when \( x = 0 \) or \( x = 9 \). So, the correct set of restrictions on \( x \) is \( \{x \in \mathbf{R} \mid x \neq 0, x \neq 9\} \). One common mistake students make in rational functions is forgetting to include all points that cause the denominator to be zero. Always check for factors in the denominator and ensure all such points are excluded from the domain of your function. Curious about real-world applications? Rational functions play a significant role in economics, representing scenarios like supply and demand ratios!