To express the transformation from $f(x) = x - 2$ to $g(x) = -2 - x$ using the $r$ notation, we need to determine how $f(x)$ is transformed into $g(x)$.

The $r$ notation typically describes reflections, rotations, translations, or scaling of functions. For reflections, we use the form $r_{y=c}$ to reflect over a horizontal line $y = c$, and $r_{x=d}$ to reflect over a vertical line $x = d$.

Since we found that the line of reflection is $y = -2$, the rule for the transformation is a reflection over this line.

Transformation Rule Using $r$ Notation

The transformation from $f(x)$ to $g(x)$ can be written as:

$g(x) = r_{y=-2}(f(x))$

This notation indicates that $g(x)$ is obtained by reflecting $f(x)$ over the line $y = -2$.

Would you like to explore more about transformations or have further questions?

Here are some related questions to consider:

1. How would you write the transformation rule using $r$ notation for a reflection over the line $x = 3$?
2. Can you express a translation of a function using $r$ notation?
3. What is the effect of reflecting a function over the line $y = x$?
4. How can you identify the transformation rule from the graph of two functions?
5. What are the different types of transformations you can represent using $r$ notation?

Tip: When working with reflections, always ensure the line of reflection is clearly identified, as it directly affects the notation and transformation rule.