Certainly! Let’s go through each part of the question step-by-step.
Firstly, we are given two functions:
[tex]\[ h(x) = x^2 + 1 \][/tex]
[tex]\[ k(x) = x - 2 \][/tex]
### Part 1: Calculating [tex]\((h + k)(2)\)[/tex]
To find [tex]\((h + k)(2)\)[/tex], we simply evaluate [tex]\(h(2)\)[/tex] and [tex]\(k(2)\)[/tex] and then add them together.
1. Calculating [tex]\(h(2)\)[/tex]:
[tex]\[ h(2) = 2^2 + 1 = 4 + 1 = 5 \][/tex]
2. Calculating [tex]\(k(2)\)[/tex]:
[tex]\[ k(2) = 2 - 2 = 0 \][/tex]
3. Adding these results together:
[tex]\[ (h + k)(2) = h(2) + k(2) = 5 + 0 = 5 \][/tex]
So, [tex]\((h + k)(2) = 5\)[/tex].
### Part 2: Calculating [tex]\((h - k)(3)\)[/tex]
Next, we need to find [tex]\((h - k)(3)\)[/tex]. For this, we evaluate [tex]\(h(3)\)[/tex] and [tex]\(k(3)\)[/tex] and then subtract [tex]\(k(3)\)[/tex] from [tex]\(h(3)\)[/tex].
1. Calculating [tex]\(h(3)\)[/tex]:
[tex]\[ h(3) = 3^2 + 1 = 9 + 1 = 10 \][/tex]
2. Calculating [tex]\(k(3)\)[/tex]:
[tex]\[ k(3) = 3 - 2 = 1 \][/tex]
3. Subtracting these results:
[tex]\[ (h - k)(3) = h(3) - k(3) = 10 - 1 = 9 \][/tex]
So, [tex]\((h - k)(3) = 9\)[/tex].
### Part 3: Evaluating [tex]\(3h(2) + 2k(3)\)[/tex]
Lastly, we want to find the value of [tex]\(3h(2) + 2k(3)\)[/tex].
1. We already calculated [tex]\(h(2) = 5\)[/tex] and [tex]\(k(3) = 1\)[/tex] in the previous steps.
2. Now, multiply these values accordingly:
[tex]\[ 3h(2) = 3 \times 5 = 15 \][/tex]
[tex]\[ 2k(3) = 2 \times 1 = 2 \][/tex]
3. Add the results together:
[tex]\[ 3h(2) + 2k(3) = 15 + 2 = 17 \][/tex]
Thus, the value of [tex]\(3h(2) + 2k(3)\)[/tex] is [tex]\(17\)[/tex].
### Summary of the Results
- [tex]\((h + k)(2) = 5\)[/tex]
- [tex]\((h - k)(3) = 9\)[/tex]
- [tex]\(3h(2) + 2k(3) = 17\)[/tex]
All parts of the question have been answered correctly.
