Answer :
To find the range of [tex]\( (w \cdot r)(x) \)[/tex] where [tex]\( r(x)=2-x^2 \)[/tex] and [tex]\( w(x)=x-2 \)[/tex], we first need to find the composite function [tex]\( (w \cdot r)(x) = w(x) \cdot r(x) \)[/tex].
1. Compose the functions:
[tex]\[ r(x) = 2 - x^2 \][/tex]
[tex]\[ w(x) = x - 2 \][/tex]
Therefore, the composite function [tex]\( (w \cdot r)(x) \)[/tex] is:
[tex]\[ (w \cdot r)(x) = (x - 2)(2 - x^2) \][/tex]
Simplifying this:
[tex]\[ (w \cdot r)(x) = (x - 2)(2 - x^2) = 2x - x^3 - 4 + 2x^2 = -x^3 + 2x^2 + 2x - 4 \][/tex]
2. Find the critical points:
To find the critical points, we need to take the derivative of [tex]\( (w \cdot r)(x) \)[/tex] and set it to zero.
[tex]\[ f(x) = -x^3 + 2x^2 + 2x - 4 \][/tex]
[tex]\[ f'(x) = \frac{d}{dx}(-x^3 + 2x^2 + 2x - 4) = -3x^2 + 4x + 2 \][/tex]
Solve [tex]\( f'(x) = 0 \)[/tex]:
[tex]\[ -3x^2 + 4x + 2 = 0 \][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = -3 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 2 \)[/tex]:
[tex]\[ x = \frac{-4 \pm \sqrt{16 + 24}}{-6} = \frac{-4 \pm \sqrt{40}}{-6} = \frac{-4 \pm 2\sqrt{10}}{-6} = \frac{2 \pm \sqrt{10}}{3} \][/tex]
So the critical points are:
[tex]\[ x = \frac{2 - \sqrt{10}}{3} \quad \text{and} \quad x = \frac{2 + \sqrt{10}}{3} \][/tex]
3. Evaluate the function at critical points:
[tex]\[ f \left( \frac{2 - \sqrt{10}}{3} \right) = \left( \frac{2 - \sqrt{10}}{3} - 2 \right) \left( 2 - \left( \frac{2 - \sqrt{10}}{3} \right)^2 \right) \][/tex]
[tex]\[ = \left( \frac{-4 - \sqrt{10}}{3} \right) \left( 2 - \frac{(2 - \sqrt{10})^2}{9} \right) \][/tex]
Simplifying these expressions (let's denote the simplified values as critical_values):
[tex]\[ f \left( \frac{2 - \sqrt{10}}{3} \right) \approx -4.41650197049510 \][/tex]
[tex]\[ f \left( \frac{2 + \sqrt{10}}{3} \right) = \left( \frac{-4 + \sqrt{10}}{3} \right) \left( 2 - \frac{(2 + \sqrt{10})^2}{9} \right) \][/tex]
[tex]\[ f \left( \frac{2 + \sqrt{10}}{3} \right) \approx 0.268353822346948 \][/tex]
4. Determine the range:
Evaluate if these critical values cover the minimum and maximum of the function:
[tex]\[ \text{Minimum value} \approx -4.41650197049510 \][/tex]
[tex]\[ \text{Maximum value} \approx 0.268353822346948 \][/tex]
Considering these values along with the domain constraints, the correct range that encompasses all possible output values of the function [tex]\( (w \cdot r)(x) \)[/tex] is:
[tex]\[ (-\infty, 2] \][/tex]
Thus, the range of [tex]\( (w \cdot r)(x) \)[/tex] is [tex]\(\boxed{(-\infty, 2]}\)[/tex].
1. Compose the functions:
[tex]\[ r(x) = 2 - x^2 \][/tex]
[tex]\[ w(x) = x - 2 \][/tex]
Therefore, the composite function [tex]\( (w \cdot r)(x) \)[/tex] is:
[tex]\[ (w \cdot r)(x) = (x - 2)(2 - x^2) \][/tex]
Simplifying this:
[tex]\[ (w \cdot r)(x) = (x - 2)(2 - x^2) = 2x - x^3 - 4 + 2x^2 = -x^3 + 2x^2 + 2x - 4 \][/tex]
2. Find the critical points:
To find the critical points, we need to take the derivative of [tex]\( (w \cdot r)(x) \)[/tex] and set it to zero.
[tex]\[ f(x) = -x^3 + 2x^2 + 2x - 4 \][/tex]
[tex]\[ f'(x) = \frac{d}{dx}(-x^3 + 2x^2 + 2x - 4) = -3x^2 + 4x + 2 \][/tex]
Solve [tex]\( f'(x) = 0 \)[/tex]:
[tex]\[ -3x^2 + 4x + 2 = 0 \][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = -3 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 2 \)[/tex]:
[tex]\[ x = \frac{-4 \pm \sqrt{16 + 24}}{-6} = \frac{-4 \pm \sqrt{40}}{-6} = \frac{-4 \pm 2\sqrt{10}}{-6} = \frac{2 \pm \sqrt{10}}{3} \][/tex]
So the critical points are:
[tex]\[ x = \frac{2 - \sqrt{10}}{3} \quad \text{and} \quad x = \frac{2 + \sqrt{10}}{3} \][/tex]
3. Evaluate the function at critical points:
[tex]\[ f \left( \frac{2 - \sqrt{10}}{3} \right) = \left( \frac{2 - \sqrt{10}}{3} - 2 \right) \left( 2 - \left( \frac{2 - \sqrt{10}}{3} \right)^2 \right) \][/tex]
[tex]\[ = \left( \frac{-4 - \sqrt{10}}{3} \right) \left( 2 - \frac{(2 - \sqrt{10})^2}{9} \right) \][/tex]
Simplifying these expressions (let's denote the simplified values as critical_values):
[tex]\[ f \left( \frac{2 - \sqrt{10}}{3} \right) \approx -4.41650197049510 \][/tex]
[tex]\[ f \left( \frac{2 + \sqrt{10}}{3} \right) = \left( \frac{-4 + \sqrt{10}}{3} \right) \left( 2 - \frac{(2 + \sqrt{10})^2}{9} \right) \][/tex]
[tex]\[ f \left( \frac{2 + \sqrt{10}}{3} \right) \approx 0.268353822346948 \][/tex]
4. Determine the range:
Evaluate if these critical values cover the minimum and maximum of the function:
[tex]\[ \text{Minimum value} \approx -4.41650197049510 \][/tex]
[tex]\[ \text{Maximum value} \approx 0.268353822346948 \][/tex]
Considering these values along with the domain constraints, the correct range that encompasses all possible output values of the function [tex]\( (w \cdot r)(x) \)[/tex] is:
[tex]\[ (-\infty, 2] \][/tex]
Thus, the range of [tex]\( (w \cdot r)(x) \)[/tex] is [tex]\(\boxed{(-\infty, 2]}\)[/tex].
