To determine the range of [tex]\((w \circ r)(x)\)[/tex], we need to understand the behavior and ranges of the individual functions [tex]\(r(x)\)[/tex] and [tex]\(w(x)\)[/tex].
### Step 1: Determine the Range of [tex]\(r(x)\)[/tex]
The function [tex]\(r(x)\)[/tex] is given by:
[tex]\[ r(x) = 2 - x^2 \][/tex]
- This is a quadratic function with a maximum value at [tex]\(x = 0\)[/tex], since the coefficient of [tex]\(x^2\)[/tex] is negative, making it a downward-opening parabola.
- At [tex]\(x = 0\)[/tex], we have [tex]\( r(0) = 2 - 0^2 = 2 \)[/tex].
As [tex]\(x \)[/tex] approaches [tex]\(\pm \infty\)[/tex], [tex]\(x^2\)[/tex] grows without bound, causing [tex]\(2 - x^2 \)[/tex] to tend towards [tex]\(-\infty\)[/tex].
Thus, the range of [tex]\(r(x)\)[/tex] is:
[tex]\[ (-\infty, 2] \][/tex]
### Step 2: Determine the Range of [tex]\(w(x)\)[/tex]
The function [tex]\(w(x)\)[/tex] is given by:
[tex]\[ w(x) = x - 2 \][/tex]
The function [tex]\(w(x)\)[/tex] is a linear function with a slope of 1. This function is defined for all real numbers [tex]\(x\)[/tex] and has no restrictions.
### Step 3: Determine the Range of the Composite Function [tex]\((w \circ r)(x)\)[/tex]
The composite function [tex]\((w \circ r)(x)\)[/tex] is:
[tex]\[ (w \circ r)(x) = w(r(x)) \][/tex]
[tex]\[ (w \circ r)(x) = r(x) - 2 \][/tex]
[tex]\[ (w \circ r)(x) = (2 - x^2) - 2 \][/tex]
[tex]\[ (w \circ r)(x) = 2 - x^2 - 2 \][/tex]
[tex]\[ (w \circ r)(x) = - x^2 \][/tex]
Now, let's determine the range of [tex]\(-x^2\)[/tex]. Since [tex]\(x^2 \geq 0\)[/tex] for all real numbers [tex]\(x\)[/tex], [tex]\(-x^2\)[/tex] will be [tex]\(\leq 0\)[/tex] for all real numbers [tex]\(x\)[/tex].
Thus, [tex]\(-x^2\)[/tex] can take on values from [tex]\(-\infty\)[/tex] to 0, inclusive. Therefore, the range of [tex]\((w \circ r)(x)\)[/tex] is:
[tex]\[ (-\infty, 0] \][/tex]
### Conclusion
The correct answer is:
[tex]\[ (-\infty, 0] \][/tex]
