Let's begin with Question 1 from the "Functions and Graphs" section:

---

Problem:

You are given two functions:
$f(x) = x - 1 \quad \text{and} \quad g(x) = \frac{2}{x-2}$

1. Find $p(x) = (f \circ g)(x)$, and determine its domain.
2. Find the inverse of $p(x)$, and sketch the graphs of $p(x)$ and $p^{-1}(x)$.

---

Solution:

1. Finding $p(x) = f(g(x))$:

First, let's substitute $g(x)$ into $f(x)$: $p(x) = f(g(x)) = f\left( \frac{2}{x-2} \right)$ Since $f(x) = x - 1$, we replace the $x$ in $f(x)$ with $g(x)$: $p(x) = \left( \frac{2}{x-2} \right) - 1$ Simplifying the expression: $p(x) = \frac{2}{x-2} - 1 = \frac{2}{x-2} - \frac{x-2}{x-2}$ $p(x) = \frac{2 - (x-2)}{x-2} = \frac{2 - x + 2}{x-2} = \frac{4 - x}{x-2}$ Thus, $p(x) = \frac{4 - x}{x-2}$.

---

Domain of $p(x)$:

• The domain of $g(x)$ is restricted by the fact that $g(x) = \frac{2}{x-2}$ is undefined when $x = 2$. Hence, $x \neq 2$.
• The expression for $p(x) = \frac{4-x}{x-2}$ does not introduce any new restrictions, so the domain remains $x \neq 2$.

Thus, the domain of $p(x)$ is:
$x \in (-\infty, 2) \cup (2, \infty)$

---

2. Finding the inverse $p^{-1}(x)$:

We now solve for $x$ in terms of $y$, where $y = p(x)$: $y = \frac{4 - x}{x - 2}$ Multiply both sides by $(x-2)$ to eliminate the denominator: $y(x - 2) = 4 - x$ Expanding both sides: $yx - 2y = 4 - x$ Rearranging the terms to isolate $x$: $yx + x = 4 + 2y$ $x(y + 1) = 4 + 2y$ Solving for $x$: $x = \frac{4 + 2y}{y + 1}$

Thus, the inverse function is: $p^{-1}(x) = \frac{4 + 2x}{x + 1}$

---

Graphing $p(x)$ and $p^{-1}(x)$:

1. Graph of $p(x) = \frac{4 - x}{x-2}$:

   • Asymptote at $x = 2$ (since $p(x)$ is undefined at $x = 2$).
   • Horizontal asymptote: As $x \to \infty$, $p(x) \to -1$, so there is a horizontal asymptote at $y = -1$.

2. Graph of $p^{-1}(x) = \frac{4 + 2x}{x + 1}$:

   • Asymptote at $x = -1$ (since $p^{-1}(x)$ is undefined at $x = -1$).
   • Horizontal asymptote: As $x \to \infty$, $p^{-1}(x) \to 2$, so there is a horizontal asymptote at $y = 2$.

Both functions are reflected over the line $y = x$.

---

Would you like more details on any part of the solution, or perhaps some assistance in visualizing the graphs?

---

Follow-up questions:

1. How do you find the composite of two functions in general?
2. What are the rules for determining the domain of a composite function?
3. How can you verify if a function has an inverse?
4. What is the relationship between the graphs of a function and its inverse?
5. How do asymptotes help in sketching graphs of rational functions?

Tip:

When finding the inverse of a function, always verify by substituting the inverse into the original function to ensure it returns the identity function $f(f^{-1}(x)) = x$.