To find and simplify [tex]\((f + g)(2)\)[/tex], we start by evaluating each function at [tex]\(x = 2\)[/tex].
First, we'll evaluate [tex]\(f(2)\)[/tex]:
[tex]\[
f(x) = -3x^2 + 3x + 3
\][/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[
f(2) = -3(2)^2 + 3(2) + 3
\][/tex]
Calculate the squares and the products:
[tex]\[
f(2) = -3(4) + 6 + 3
\][/tex]
[tex]\[
f(2) = -12 + 6 + 3
\][/tex]
Combine the constants:
[tex]\[
f(2) = -12 + 9 = -3
\][/tex]
Next, we'll evaluate [tex]\(g(2)\)[/tex]:
[tex]\[
g(x) = 3x - 4
\][/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[
g(2) = 3(2) - 4
\][/tex]
Calculate the product:
[tex]\[
g(2) = 6 - 4
\][/tex]
Combine the constants:
[tex]\[
g(2) = 2
\][/tex]
Now, to find [tex]\((f + g)(2)\)[/tex], we add the values of [tex]\(f(2)\)[/tex] and [tex]\(g(2)\)[/tex]:
[tex]\[
(f + g)(2) = f(2) + g(2)
\][/tex]
Substitute the values we found:
[tex]\[
(f + g)(2) = -3 + 2
\][/tex]
Combine the constants:
[tex]\[
(f + g)(2) = -1
\][/tex]
Thus, the simplified value of [tex]\((f + g)(2)\)[/tex] is [tex]\(-1\)[/tex].
