The function provided is:

$f(x) = \sqrt{x - 2} - 4$

Step 1: Identify the Basic Function

The core of this function is the square root, $\sqrt{x}$. The more basic function is:

$g(x) = \sqrt{x}$

Now, let's analyze the transformations applied to this basic function to get $f(x)$.

Transformations:

1. Horizontal Shift:

   • Inside the square root, we have $x - 2$. This represents a horizontal shift to the right by 2 units.
   • The transformation is $g(x) = \sqrt{x - 2}$.

2. Vertical Shift:

   • The constant $-4$ outside the square root represents a vertical shift downward by 4 units.
   • The transformation is $f(x) = \sqrt{x - 2} - 4$.

Thus, the basic function $g(x) = \sqrt{x}$ has been shifted to the right by 2 units and shifted downward by 4 units to obtain $f(x)$.

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Would you like more details on these transformations, or do you have any specific questions?

Here are 5 related questions to expand on this topic:

1. What is the domain of the function $f(x)$?
2. How would you reflect this function across the x-axis?
3. What effect does multiplying the square root by a constant have?
4. How would the graph change if the function was $f(x) = -\sqrt{x - 2} - 4$?
5. What would the function look like if the vertical shift was +4 instead of -4?

Tip: A horizontal shift is always the opposite of the sign inside the function. For example, $x - 2$ means shifting right by 2, not left!