To determine which of these functions defines a one-to-one correspondence (bijection) from the domain $(0, 2)$ to the range $(3, \infty)$, we analyze the behavior of each function.

Step 1: Criteria for One-to-One Correspondence

• Injective (One-to-One): Each $x$ in the domain maps to a unique $y$ in the range.
• Surjective (Onto): Every value in the range $(3, \infty)$ must be reachable from some $x$ in the domain $(0, 2)$.

Step 2: Analyze Each Function

A) $f(x) = \frac{2}{x} + 2$

1. Domain Check: For $x \in (0, 2)$, $f(x)$ is defined.
2. Range Check:
   • As $x \to 0^+$, $\frac{2}{x} \to \infty$, so $f(x) \to \infty$.
   • As $x \to 2^-$, $\frac{2}{x} \to 1$, so $f(x) \to 3$.
   • Range: $(3, \infty)$.
3. $f(x)$ is strictly decreasing (derivative is negative), ensuring injectivity.

This function is a bijection.

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B) $f(x) = \frac{3}{x} + 3$

1. Domain Check: For $x \in (0, 2)$, $f(x)$ is defined.
2. Range Check:
   • As $x \to 0^+$, $\frac{3}{x} \to \infty$, so $f(x) \to \infty$.
   • As $x \to 2^-$, $\frac{3}{x} \to \frac{3}{2}$, so $f(x) \to \frac{3}{2} + 3 = 4.5$.
   • Range: $(4.5, \infty)$.
3. Since $(4.5, \infty) \neq (3, \infty)$, this is not surjective.

Not a bijection.

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C) $f(x) = \frac{1}{x} + 2$

1. Domain Check: For $x \in (0, 2)$, $f(x)$ is defined.
2. Range Check:
   • As $x \to 0^+$, $\frac{1}{x} \to \infty$, so $f(x) \to \infty$.
   • As $x \to 2^-$, $\frac{1}{x} \to \frac{1}{2}$, so $f(x) \to 2.5$.
   • Range: $(2.5, \infty)$.
3. Since $(2.5, \infty) \neq (3, \infty)$, this is not surjective.

Not a bijection.

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D) $f(x) = \frac{1}{x} + 3$

1. Domain Check: For $x \in (0, 2)$, $f(x)$ is defined.
2. Range Check:
   • As $x \to 0^+$, $\frac{1}{x} \to \infty$, so $f(x) \to \infty$.
   • As $x \to 2^-$, $\frac{1}{x} \to \frac{1}{2}$, so $f(x) \to 3.5$.
   • Range: $(3.5, \infty)$.
3. Since $(3.5, \infty) \neq (3, \infty)$, this is not surjective.

Not a bijection.

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E) None of the mentioned

Since $f(x) = \frac{2}{x} + 2$ satisfies the conditions for a bijection from $(0, 2)$ to $(3, \infty)$, this option is incorrect.

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Final Answer:

A) $f(x) = \frac{2}{x} + 2$

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Do you want a detailed explanation for any specific step?

Here are 5 related questions to explore:

1. How do you verify if a function is strictly increasing or decreasing?
2. What are the conditions for a function to be bijective?
3. Can you find the derivative of $f(x) = \frac{2}{x} + 2$ and interpret it?
4. Why do functions like $f(x) = \frac{1}{x} + 3$ fail to map onto $(3, \infty)$?
5. What changes in the function domain or codomain would make $f(x) = \frac{1}{x} + 3$ bijective?

Tip: Always test the limits of a function at the boundaries of its domain to verify its range.