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.