To find the **domain** and **range** of the inverse function \( f^{-1}(x) = x^2 - 2x + 6 \), let's carefully analyze it step by step.

### 1. **Domain of \( f^{-1}(x) \)**:
The domain of a function refers to the set of all possible input values (i.e., \( x \)) for which the function is defined. In this case, \( f^{-1}(x) = x^2 - 2x + 6 \) is a quadratic function, which is defined for all real numbers.

- **Domain**: \( (-\infty, \infty) \)

### 2. **Range of \( f^{-1}(x) \)**:
The range refers to the set of all possible output values of the function. Since \( f^{-1}(x) \) is a quadratic function with a positive leading coefficient (the coefficient of \( x^2 \) is 1), the graph of the function is a **parabola opening upwards**. 

We can find the minimum value of the quadratic function by completing the square or using the vertex formula.

The vertex of a parabola \( ax^2 + bx + c \) occurs at:
\[
x = -\frac{b}{2a} = -\frac{-2}{2(1)} = 1
\]
Substitute \( x = 1 \) into the function to find the corresponding \( y \)-value:
\[
f^{-1}(1) = (1)^2 - 2(1) + 6 = 1 - 2 + 6 = 5
\]
Thus, the minimum value of \( f^{-1}(x) \) is 5, and since the parabola opens upwards, the range is from 5 to infinity.

- **Range**: \( [5, \infty) \)

### Summary:
- **Domain** of \( f^{-1}(x) \): \( (-\infty, \infty) \)
- **Range** of \( f^{-1}(x) \): \( [5, \infty) \)

Would you like any further details on this explanation?

### Related Questions:
1. What are the steps to graph a quadratic function like \( f^{-1}(x) = x^2 - 2x + 6 \)?
2. How do you find the inverse of a function step by step?
3. How does completing the square help in finding the vertex of a quadratic function?
4. Can the range of a quadratic function ever be negative?
5. What are the domain and range of the original function \( f(x) \), if we are given \( f^{-1}(x) \)?

### Tip:
For any quadratic function of the form \( ax^2 + bx + c \), the minimum or maximum value occurs at the vertex, given by \( x = -\frac{b}{2a} \).

Find the domain and range of the inverse function f^(-1)(x) = x^2 - 2x + 6. This solution explains the process of determining the domain and finding the vertex to identify the range of the quadratic function, with a clear step-by-step guide for high school students.

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