To determine the range of the composition of the functions [tex]\(r(x)\)[/tex] and [tex]\(w(x)\)[/tex], let's follow these steps:
1. Define the functions:
- [tex]\( r(x) = 2 - x^2 \)[/tex]
- [tex]\( w(x) = x - 2 \)[/tex]
2. Find the composition [tex]\( (w \circ r)(x) \)[/tex], which means [tex]\( w(r(x)) \)[/tex].
Substitute [tex]\( r(x) \)[/tex] into [tex]\( w(x) \)[/tex]:
[tex]\[
w(r(x)) = w(2 - x^2)
\][/tex]
3. Evaluate [tex]\( w(2 - x^2) \)[/tex]:
- [tex]\( w(2 - x^2) = (2 - x^2) - 2 = -x^2 \)[/tex]
4. Determine the range of [tex]\( -x^2 \)[/tex]:
- Consider the function [tex]\( -x^2 \)[/tex].
- The function [tex]\( x^2 \)[/tex] is always non-negative (i.e., [tex]\(x^2 \geq 0\)[/tex]), and thus [tex]\( x^2 \)[/tex] takes values in the range [tex]\([0, \infty)\)[/tex].
- Multiplying by [tex]\(-1\)[/tex], [tex]\( -x^2 \)[/tex] will take values in the range [tex]\((- \infty, 0]\)[/tex].
Thus, the range of [tex]\( (w \circ r)(x) \)[/tex] is:
[tex]\[
(-\infty, 0]
\][/tex]
Therefore, the correct answer is:
[tex]\[
(-\infty, 0]
\][/tex]
