Let's solve each expression step by step to find their respective values.
1. Evaluate [tex]\( f(2) \)[/tex]:
[tex]\[
f(x) = 3x^2 + 2
\][/tex]
Substitute [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = 3(2)^2 + 2 = 3(4) + 2 = 12 + 2 = 14
\][/tex]
2. Evaluate [tex]\( g(3) \)[/tex]:
[tex]\[
g(x) = 4x^2 + 2x - 1
\][/tex]
Substitute [tex]\( x = 3 \)[/tex]:
[tex]\[
g(3) = 4(3)^2 + 2(3) - 1 = 4(9) + 6 - 1 = 36 + 6 - 1 = 41
\][/tex]
3. Evaluate [tex]\( h(4) \)[/tex]:
[tex]\[
h(x) = 2x - 3
\][/tex]
Substitute [tex]\( x = 4 \)[/tex]:
[tex]\[
h(4) = 2(4) - 3 = 8 - 3 = 5
\][/tex]
4. Evaluate [tex]\( f(1) + g(1) \)[/tex]:
[tex]\[
f(x) = 3x^2 + 2,\quad g(x) = 4x^2 + 2x - 1
\][/tex]
Substitute [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 3(1)^2 + 2 = 3(1) + 2 = 3 + 2 = 5
\][/tex]
[tex]\[
g(1) = 4(1)^2 + 2(1) - 1 = 4(1) + 2(1) - 1 = 4 + 2 - 1 = 5
\][/tex]
[tex]\[
f(1) + g(1) = 5 + 5 = 10
\][/tex]
5. Evaluate [tex]\( h(2) + f(2) \)[/tex]:
[tex]\[
h(x) = 2x - 3,\quad f(x) = 3x^2 + 2
\][/tex]
Substitute [tex]\( x = 2 \)[/tex]:
[tex]\[
h(2) = 2(2) - 3 = 4 - 3 = 1
\][/tex]
[tex]\[
f(2) = 3(2)^2 + 2 = 3(4) + 2 = 12 + 2 = 14
\][/tex]
[tex]\[
h(2) + f(2) = 1 + 14 = 15
\][/tex]
Finally, let's match the expressions with their results:
[tex]\[
\begin{array}{l}
f(2) = 14 \\
g(3) = 41 \\
h(4) = 5 \\
f(1) + g(1) = 10 \\
h(2) + f(2) = 15 \\
\end{array}
\][/tex]
Thus, the correct answers to the given functions are:
[tex]\[
\begin{array}{l}
f(2) = 14 \\
g(3) = 41 \\
h(4) = 5 \\
f(1) + g(1) = 10 \\
h(2) + f(2) = 15 \\
\end{array}
\][/tex]
