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We are given the following problem:

Let \( f \) be a function such that \( f \) and \( f^{-1} \) are defined on \( \mathbb{R} \), and \( f(x) \) is a one-to-one function. We also know that:
\[
2 + 2f(2) = 6
\]
and we need to find \( f^{-1}(2) \).

### Step 1: Solve for \( f(2) \)
The equation provided is:
\[
2 + 2f(2) = 6
\]
Subtracting 2 from both sides:
\[
2f(2) = 4
\]
Dividing both sides by 2:
\[
f(2) = 2
\]

### Step 2: Interpret \( f^{-1}(2) \)
By definition of the inverse function, if \( f(a) = b \), then \( f^{-1}(b) = a \).

Since we just found that \( f(2) = 2 \), this means:
\[
f^{-1}(2) = 2
\]

### Final Answer:
\( f^{-1}(2) = 2 \), so the correct option is \( \boxed{2} \).

Would you like more details or have any questions?

### Related Questions:
1. What is a one-to-one function?
2. How do you find the inverse of a function algebraically?
3. Can a function have more than one inverse?
4. What are the conditions for a function to have an inverse?
5. How do transformations of functions affect their inverses?

**Tip:** Always remember that a function must be one-to-one to have an inverse, meaning each output is associated with exactly one input.

Let f be a function such that f and f⁻¹ are defined on ℝ, if f(x) is a (1-1) function with 2 + 2f(2) = 6, then f⁻¹(2) = ?

This problem involves finding the inverse of a one-to-one function f(x), defined by the equation 2 + 2f(2) = 6. We solve for f(2) and use the definition of inverse functions to determine f⁻¹(2).

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