Let's evaluate the given expressions step-by-step:
### 1. Define and evaluate the functions [tex]\( h(x) \)[/tex] and [tex]\( k(x) \)[/tex]:
Given function [tex]\( h(x) \)[/tex]:
[tex]\[ h(x) = x^2 + 1 \][/tex]
Let's evaluate [tex]\( h(2) \)[/tex]:
[tex]\[ h(2) = 2^2 + 1 = 4 + 1 = 5 \][/tex]
Given function [tex]\( k(x) \)[/tex]:
[tex]\[ k(x) = x + 1 \][/tex]
Let's evaluate [tex]\( k(2) \)[/tex]:
[tex]\[ k(2) = 2 + 1 = 3 \][/tex]
Let's evaluate [tex]\( k(3) \)[/tex]:
[tex]\[ k(3) = 3 + 1 = 4 \][/tex]
### 2. Compute [tex]\((h + k)(2)\)[/tex]:
[tex]\[ (h + k)(2) = h(2) + k(2) = 5 + 3 = 8 \][/tex]
### 3. Compute [tex]\((h - k)(3)\)[/tex]:
[tex]\[ (h - k)(3) = h(3) - k(3) \][/tex]
Let's first find [tex]\( h(3) \)[/tex]:
[tex]\[ h(3) = 3^2 + 1 = 9 + 1 = 10 \][/tex]
Now calculate [tex]\((h - k)(3)\)[/tex]:
[tex]\[ (h - k)(3) = h(3) - k(3) = 10 - 4 = 6 \][/tex]
### 4. Evaluate the expression [tex]\( 3h(2) + 2k(3) \)[/tex]:
Now, using the previously computed values:
[tex]\[ 3h(2) + 2k(3) = 3 \times 5 + 2 \times 4 \][/tex]
[tex]\[ 3h(2) + 2k(3) = 15 + 8 = 23 \][/tex]
### Final Results:
[tex]\[
\begin{array}{l}
(h+k)(2) = 8 \\
(h-k)(3) = 6 \\
3h(2) + 2k(3) = 23
\end{array}
\][/tex]
Thus, the values are:
[tex]\[
\boxed{8}
\boxed{6}
\boxed{23}
\][/tex]
