Let's work through this problem to fill in the missing steps. The function given is:

$f : (1, \infty) \to (1, \infty), \quad f(x) = \frac{x}{x - 1}.$

The task involves proving that this function is one-to-one (injective) and onto (surjective).

Part (a) Proving Injectivity (One-to-One)

To show that $f$ is one-to-one, assume $f(a) = f(b)$ for some $a, b \in (1, \infty)$ and show that this implies $a = b$.

1. We start with: $f(a) = f(b) \Rightarrow \frac{a}{a - 1} = \frac{b}{b - 1}.$

2. Cross-multiplying to clear the denominators, we get: $a(b - 1) = b(a - 1).$

3. Expanding both sides: $ab - a = ba - b.$

4. Subtracting $ab$ from both sides: $-a = -b.$

5. Multiplying both sides by $-1$: $a = b.$

Therefore, $f$ is one-to-one.

Part (b) Proving Surjectivity (Onto)

To show that $f$ is onto, let $b$ be an element in the codomain $(1, \infty)$. We need to show that there exists an $a$ in the domain $(1, \infty)$ such that $f(a) = b$.

1. Suppose $f(a) = b$, then: $\frac{a}{a - 1} = b.$

2. Multiply both sides by $a - 1$: $a = b(a - 1).$

3. Rearrange to isolate terms involving $a$ on one side: $a - ba = -b.$

4. Factor out $a$: $a(1 - b) = -b.$

5. Divide by $1 - b$ (note that $b > 1$, so $1 - b < 0$): $a = \frac{b}{b - 1}.$

Thus, for each $b \in (1, \infty)$, there exists an $a \in (1, \infty)$ such that $f(a) = b$, proving that $f$ is onto.

Since $f$ is both one-to-one and onto, it is invertible.

Finding the Inverse $f^{-1}(x)$

To find $f^{-1}(x)$, set $y = f(x)$ and solve for $x$ in terms of $y$:

1. Let $y = \frac{x}{x - 1}$.
2. Multiply by $x - 1$: $y(x - 1) = x.$
3. Expand and rearrange to isolate $x$: $yx - y = x.$
4. Collect $x$-terms on one side: $x(y - 1) = y.$
5. Solve for $x$: $x = \frac{y}{y - 1}.$

Therefore, the inverse function is: $f^{-1}(x) = \frac{x}{x - 1}.$

Summary

1. Injective: $f(a) = f(b) \Rightarrow a = b$.
2. Surjective: For every $b \in (1, \infty)$, there exists $a \in (1, \infty)$ such that $f(a) = b$.
3. Inverse function: $f^{-1}(x) = \frac{x}{x - 1}$.

Would you like further explanation on any of these steps, or do you have any specific questions?

Additional Questions for Practice

1. What is the general method to prove a function is injective?
2. How can you verify that the inverse function $f^{-1}(x) = \frac{x}{x - 1}$ is correct?
3. Why is the function $f(x) = \frac{x}{x - 1}$ restricted to the domain $(1, \infty)$?
4. How would the proof of surjectivity change if the codomain were different?
5. Can you apply this method of finding the inverse to other rational functions?

Tip

When proving a function is injective, always start by assuming $f(a) = f(b)$ and try to derive $a = b$. This approach is foundational in function analysis.