To solve these problems, let's evaluate each function and pair them appropriately.
1. Evaluate [tex]\((g - f)(-1)\)[/tex]:
For [tex]\(g(x) = \sqrt{11 - 4x}\)[/tex]:
[tex]\[
g(-1) = \sqrt{11 - 4(-1)} = \sqrt{11 + 4} = \sqrt{15}
\][/tex]
For [tex]\(f(x) = 1 - x^2\)[/tex]:
[tex]\[
f(-1) = 1 - (-1)^2 = 1 - 1 = 0
\][/tex]
Therefore:
[tex]\[
(g - f)(-1) = g(-1) - f(-1) = \sqrt{15} - 0 = \sqrt{15}
\][/tex]
2. Evaluate [tex]\((g \times f)(2)\)[/tex]:
For [tex]\(g(x) = \sqrt{11 - 4x}\)[/tex]:
[tex]\[
g(2) = \sqrt{11 - 4(2)} = \sqrt{11 - 8} = \sqrt{3}
\][/tex]
For [tex]\(f(x) = 1 - x^2\)[/tex]:
[tex]\[
f(2) = 1 - 2^2 = 1 - 4 = -3
\][/tex]
Therefore:
[tex]\[
(g \times f)(2) = g(2) \times f(2) = \sqrt{3} \times (-3) = -3\sqrt{3}
\][/tex]
3. Evaluate [tex]\(\left(\frac{f}{g}\right)(-1)\)[/tex]:
For [tex]\(g(x) = \sqrt{11 - 4x}\)[/tex]:
[tex]\[
g(-1) = \sqrt{11 - 4(-1)} = \sqrt{11 + 4} = \sqrt{15}
\][/tex]
For [tex]\(f(x) = 1 - x^2\)[/tex]:
[tex]\[
f(-1) = 1 - (-1)^2 = 1 - 1 = 0
\][/tex]
Therefore:
[tex]\[
\left(\frac{f}{g}\right)(-1) = \frac{f(-1)}{g(-1)} = \frac{0}{\sqrt{15}} = 0
\][/tex]
4. Evaluate [tex]\((g + f)(2)\)[/tex]:
For [tex]\(g(x) = \sqrt{11 - 4x}\)[/tex]:
[tex]\[
g(2) = \sqrt{11 - 4(2)} = \sqrt{11 - 8} = \sqrt{3}
\][/tex]
For [tex]\(f(x) = 1 - x^2\)[/tex]:
[tex]\[
f(2) = 1 - 2^2 = 1 - 4 = -3
\][/tex]
Therefore:
[tex]\[
(g + f)(2) = g(2) + f(2) = \sqrt{3} + (-3) = \sqrt{3} - 3
\][/tex]
So, the results for the given operations are:
1. [tex]\((g - f)(-1) = 3.872983346207417\)[/tex]
2. [tex]\((g \times f)(2) = -5.196152422706632\)[/tex]
3. [tex]\(\left(\frac{f}{g}\right)(-1) = 0.0\)[/tex]
4. [tex]\((g + f)(2) = -1.2679491924311228\)[/tex]
These results match perfectly with the answers we have:
- [tex]\((g - f)(-1)\)[/tex] corresponds to [tex]\(3.872983346207417\)[/tex]
- [tex]\((g \times f)(2)\)[/tex] corresponds to [tex]\(-5.196152422706632\)[/tex]
- [tex]\(\left(\frac{f}{g}\right)(-1)\)[/tex] corresponds to [tex]\(0.0\)[/tex]
- [tex]\((g + f)(2)\)[/tex] corresponds to [tex]\(-1.2679491924311228\)[/tex]
Thus, we have the correct pairs:
- (g - f)(-1) ⟶ [tex]\(3.872983346207417\)[/tex]
- (g × f)(2) ⟶ [tex]\(-5.196152422706632\)[/tex]
- [tex]\(\left(\frac{f}{g}\right)(-1)\)[/tex] ⟶ [tex]\(0.0\)[/tex]
- (g + f)(2) ⟶ [tex]\(-1.2679491924311228\)[/tex]
