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Let's analyze the problem.

### Given Function:
\[ f(x) = \sqrt{x - 6} \]

#### Part (a) - Find the inverse function \( f^{-1}(x) \).

1. Start with:
   \[
   y = f(x) = \sqrt{x - 6}
   \]
2. Switch \(x\) and \(y\) to find the inverse:
   \[
   x = \sqrt{y - 6}
   \]
3. Square both sides to eliminate the square root:
   \[
   x^2 = y - 6
   \]
4. Solve for \(y\):
   \[
   y = x^2 + 6
   \]
Thus, the inverse function is:
\[
f^{-1}(x) = x^2 + 6
\]

#### Part (b) - Graph of \( f(x) \) and \( f^{-1}(x) \)

The graph of \( f(x) = \sqrt{x - 6} \) is a square root function, which starts at \(x = 6\) and increases as \(x\) increases. The graph of \( f^{-1}(x) = x^2 + 6 \) is a parabola that opens upwards with its vertex at \( (0, 6) \).

#### Part (c) - Domain and Range of \( f(x) \) and \( f^{-1}(x) \)

- The domain of \( f(x) = \sqrt{x - 6} \) is \( [6, \infty) \), since the expression under the square root must be non-negative.
- The range of \( f(x) \) is \( [0, \infty) \), since square root functions yield non-negative outputs.
- The domain of \( f^{-1}(x) = x^2 + 6 \) is \( (-\infty, \infty) \), since squaring any real number is valid.
- The range of \( f^{-1}(x) \) is \( [6, \infty) \), as the lowest value occurs when \(x = 0\), yielding 6.

#### Question Selection (From Image):
The inverse function \( f^{-1}(x) \) is:
\[
f^{-1}(x) = x^2 + 6
\]
This implies that the correct option from the choices given is:
- Option C: \( f^{-1}(x) = x^2 + 6 \), for all \(x\).

Would you like further clarification or details on any part of this? 

### Related Questions:
1. What are the properties of inverse functions in general?
2. How would the graph of \(f(x)\) change if the function was \( f(x) = \sqrt{x - 2} \)?
3. Can you explain the process of graphing both \( f(x) \) and \( f^{-1}(x) \)?
4. What role does the square root function play in determining the domain of \( f(x) \)?
5. What happens to the range if we apply transformations to \( f(x) \)?

**Tip:** The domain of a function is always the range of its inverse, and vice versa.

Given the function f(x) = √(x - 6), find the inverse function f⁻¹(x). Then, use the inverse function to complete the answer choices.

Learn how to find the inverse function for f(x) = √(x - 6) and determine the correct domain and range. This solution includes steps for calculating the inverse function and applying the square root property, suitable for students in Grades 10-12.

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