The following questions are open-ended, meaning, there may or may not be more than one answer.
6. Create an equation for a rational function whose graph is the line $y=x$ with a single hole. Explain your reasoning.
(2 marks)
7. Write an equation for a rational function whose graph has all the following features: (3 marks)
- vertical asymptote with equation $x=3$
- horizontal asymptote with equation $y=2$
- hole at $x=1$
8. Write an equation for a rational function whose graph has no $x$-intercepts and a vertical asymptote with equation $x=2$. Explain your reasoning.
(2 marks)

Step 1 ：Let the rational function be \(\frac{f(x)}{g(x)}\), where \(f(x)\) and \(g(x)\) are polynomials.

Step 2 ：To create a hole at \(x=1\), introduce a common factor in the numerator and denominator that will cancel out at this specific point. Let this factor be \((x-1)\).

Step 3 ：Since the graph is the line \(y=x\) for all other values of \(x\), we can multiply the factor \((x-1)\) by \(x\) in the numerator to obtain the function \(f(x) = x(x-1)\).

Step 4 ：Thus, the rational function is \(\frac{x(x-1)}{x-1}\).

Step 5 ：Final Answer: The equation for a rational function whose graph is the line \(y=x\) with a single hole is \(\boxed{\frac{x(x-1)}{x-1}}\). The hole occurs at \(x=1\) because the factor \((x-1)\) cancels out in the numerator and denominator, leaving \(y=x\) for all other values of \(x\).