The problem you're referring to from the image involves a function $f: A \to B$ defined as $f(x) = \frac{4x}{x-2}$. It asks to determine the domain $A$ and codomain $B$ so that the function is bijective and has an inverse.

Let's break this down into steps:

1. Determine the Domain: The function is defined as $f(x) = \frac{4x}{x-2}$, which is undefined when the denominator is zero. Therefore, we need to exclude the value $x = 2$ from the domain.

   $A = \{x \in \mathbb{R} \mid x \neq 2 \}$

2. Determine Bijectivity: A function is bijective if it is both injective (one-to-one) and surjective (onto).

   • Injective: To check injectivity, we need to show that if $f(x_1) = f(x_2)$, then $x_1 = x_2$. Let's check this algebraically: $\frac{4x_1}{x_1-2} = \frac{4x_2}{x_2-2}$ Cross-multiplying and simplifying leads to $x_1 = x_2$, proving injectivity.

   • Surjective: The function must map every element in $B$ (the codomain) to some element in $A$ (the domain). To find the range, we solve for $y$ in terms of $x$: $y = \frac{4x}{x-2}$ Solve for $x$ in terms of $y$: $y(x - 2) = 4x \implies yx - 2y = 4x \implies yx - 4x = 2y \implies x(y - 4) = 2y$ $x = \frac{2y}{y - 4}$ So, the function is surjective as long as $y \neq 4$. Hence, the codomain $B$ must exclude 4: $B = \{ y \in \mathbb{R} \mid y \neq 4 \}$

3. Find the Inverse: The inverse function is found by solving $y = \frac{4x}{x-2}$ for $x$: $x = \frac{2y}{y - 4}$ Thus, the inverse function is: $f^{-1}(y) = \frac{2y}{y - 4}$

Summary:

• The domain $A$ is $\mathbb{R} \setminus \{2\}$,
• The codomain $B$ is $\mathbb{R} \setminus \{4\}$,
• The inverse function is $f^{-1}(y) = \frac{2y}{y - 4}$.

Would you like further details on any part of this explanation?

Here are 5 follow-up questions:

1. Why is injectivity important for a function to have an inverse?
2. How can you geometrically interpret bijective functions?
3. What happens if the domain or codomain is incorrectly specified?
4. How can you generalize the process of finding an inverse for rational functions?
5. What role does asymptotic behavior play in defining the domain and range?

Tip: When finding the inverse of a function, ensure that the original function is bijective, otherwise, the inverse might not exist!