Let's start by writing down the functions and the values we need to evaluate:
### Given:
- [tex]\( h(x) = x^2 + 1 \)[/tex]
- [tex]\( k(x) = x - 2 \)[/tex]
### Evaluating [tex]\((h+k)(2)\)[/tex]:
[tex]\[
(h+k)(x) = h(x) + k(x)
\][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[
(h+k)(2) = h(2) + k(2)
\][/tex]
Next, we individually evaluate [tex]\( h(2) \)[/tex] and [tex]\( k(2) \)[/tex]:
[tex]\[
h(2) = 2^2 + 1 = 4 + 1 = 5
\][/tex]
[tex]\[
k(2) = 2 - 2 = 0
\][/tex]
So:
[tex]\[
(h+k)(2) = h(2) + k(2) = 5 + 0 = 5
\][/tex]
### Evaluating [tex]\((h-k)(3)\)[/tex]:
[tex]\[
(h-k)(x) = h(x) - k(x)
\][/tex]
For [tex]\( x = 3 \)[/tex]:
[tex]\[
(h-k)(3) = h(3) - k(3)
\][/tex]
Next, we individually evaluate [tex]\( h(3) \)[/tex] and [tex]\( k(3) \)[/tex]:
[tex]\[
h(3) = 3^2 + 1 = 9 + 1 = 10
\][/tex]
[tex]\[
k(3) = 3 - 2 = 1
\][/tex]
So:
[tex]\[
(h-k)(3) = h(3) - k(3) = 10 - 1 = 9
\][/tex]
### Evaluating [tex]\( 3 h(2) + 2 k(3) \)[/tex]:
We already found [tex]\( h(2) \)[/tex] and [tex]\( k(3) \)[/tex]:
[tex]\[
h(2) = 5
\][/tex]
[tex]\[
k(3) = 1
\][/tex]
Now substitute these values into the expression:
[tex]\[
3 h(2) + 2 k(3) = 3 \cdot 5 + 2 \cdot 1 = 15 + 2 = 17
\][/tex]
### Summary:
The evaluated values are:
[tex]\[
(h+k)(2) = 5
\][/tex]
[tex]\[
(h-k)(3) = 9
\][/tex]
[tex]\[
3 h(2) + 2 k(3) = 17
\][/tex]
Putting it all together:
1. [tex]\((h+k)(2) = 5\)[/tex]
2. [tex]\((h-k)(3) = 9\)[/tex]
3. [tex]\(3 h(2) + 2 k(3) = 17\)[/tex]
