To evaluate each combined function and match it to the corresponding value, let's look at each function step-by-step:
1. Evaluate [tex]\(\left(\frac{f}{g}\right)(-1)\)[/tex]:
[tex]$ f(x) = 1 - x^2 $[/tex]
[tex]$ f(-1) = 1 - (-1)^2 = 1 - 1 = 0 $[/tex]
[tex]$ g(x) = \sqrt{11 - 4x} $[/tex]
[tex]$ g(-1) = \sqrt{11 - 4(-1)} = \sqrt{11 + 4} = \sqrt{15} $[/tex]
[tex]$ \left(\frac{f}{g}\right)(-1) = \frac{f(-1)}{g(-1)} = \frac{0}{\sqrt{15}} = 0 $[/tex]
The combined function [tex]\(\left(\frac{f}{g}\right)(-1)\)[/tex] corresponds to [tex]\(0\)[/tex].
2. Evaluate [tex]\((g + f)(2)\)[/tex]:
[tex]$ f(2) = 1 - 2^2 = 1 - 4 = -3 $[/tex]
[tex]$ g(2) = \sqrt{11 - 4(2)} = \sqrt{11 - 8} = \sqrt{3} $[/tex]
[tex]$ (g + f)(2) = g(2) + f(2) = \sqrt{3} - 3 $[/tex]
The combined function [tex]\((g + f)(2)\)[/tex] corresponds to [tex]\(\sqrt{3} - 3\)[/tex].
3. Evaluate [tex]\((g - f)(-1)\)[/tex]:
We've already found [tex]\( f(-1) = 0 \)[/tex] and [tex]\( g(-1) = \sqrt{15} \)[/tex].
[tex]$ (g - f)(-1) = g(-1) - f(-1) = \sqrt{15} - 0 = \sqrt{15} $[/tex]
The combined function [tex]\((g - f)(-1)\)[/tex] corresponds to [tex]\(\sqrt{15}\)[/tex].
4. Evaluate [tex]\((g \cdot f)(2)\)[/tex]:
We've already found [tex]\( f(2) = -3 \)[/tex] and [tex]\( g(2) = \sqrt{3} \)[/tex].
[tex]$ (g \cdot f)(2) = g(2) \cdot f(2) = \sqrt{3} \cdot (-3) = -3\sqrt{3} $[/tex]
The combined function [tex]\((g \cdot f)(2)\)[/tex] corresponds to [tex]\(-3\sqrt{3}\)[/tex].
Putting it all together, we get the following pairs:
[tex]\[
\begin{array}{ll}
\left(\frac{f}{g}\right)(-1) & \longrightarrow 0 \\
(g + f)(2) & \longrightarrow \sqrt{3} - 3 \\
(g - f)(-1) & \longrightarrow \sqrt{15} \\
(g \cdot f)(2) & \longrightarrow -3\sqrt{3} \\
\end{array}
\][/tex]
