Alright, let's solve this problem step-by-step.
We are given two functions:
[tex]\[
f(x) = 1 - x^2
\][/tex]
and
[tex]\[
g(x) = \sqrt{11 - 4x}
\][/tex]
Let's calculate the needed operations and match them to their respective answers:
### 1. Calculate [tex]\((g-f)(-1)\)[/tex]
First, find [tex]\(f(-1)\)[/tex] and [tex]\( g(-1) \)[/tex]:
[tex]\[ f(-1) = 1 - (-1)^2 = 1 - 1 = 0 \][/tex]
[tex]\[ g(-1) = \sqrt{11 - 4(-1)} = \sqrt{11 + 4} = \sqrt{15} \][/tex]
Now, calculate [tex]\((g - f)(-1)\)[/tex]:
[tex]\[ (g - f)(-1) = g(-1) - f(-1) = \sqrt{15} - 0 = \sqrt{15} \][/tex]
### 2. Calculate [tex]\((g + f)(2)\)[/tex]
First, find [tex]\(f(2)\)[/tex] and [tex]\(g(2)\)[/tex]:
[tex]\[ f(2) = 1 - 2^2 = 1 - 4 = -3 \][/tex]
[tex]\[ g(2) = \sqrt{11 - 4 \cdot 2} = \sqrt{11 - 8} = \sqrt{3} \][/tex]
Now, calculate [tex]\((g + f)(2)\)[/tex]:
[tex]\[ (g + f)(2) = g(2) + f(2) = \sqrt{3} - 3 \][/tex]
### 3. Calculate [tex]\(\left(\frac{f}{g}\right)(-1)\)[/tex]
Use the values we previously found:
[tex]\[ \left(\frac{f}{g}\right)(-1) = \frac{f(-1)}{g(-1)} = \frac{0}{\sqrt{15}} = 0 \][/tex]
### 4. Match [tex]\(\sqrt{3}-3 \rightarrow \square\)[/tex]
From the above calculations, we see that:
[tex]\[
(g+f)(2) = \sqrt{3} - 3
\][/tex]
So, the correct matching of tiles should be:
1. [tex]\((g-f)(-1) = \sqrt{15}\)[/tex]
2. [tex]\((g+f)(2) = \sqrt{3} - 3\)[/tex]
3. [tex]\(\left(\frac{f}{g}\right)(-1) = 0\)[/tex]
4. [tex]\(\sqrt{3} - 3 \rightarrow (g + f)(2) \)[/tex]
Therefore, the completed pairs are:
[tex]\[
(g - f)(-1) \rightarrow \sqrt{15}
\][/tex]
[tex]\[
(g + f)(2) \rightarrow \sqrt{3} - 3
\][/tex]
[tex]\[
\left(\frac{f}{g}\right)(-1) \rightarrow 0
\][/tex]
[tex]\[
\sqrt{3}-3 \rightarrow (g + f)(2)
\][/tex]
