Certainly! Let's go through each part step-by-step.
First, the functions [tex]\( h(x) \)[/tex] and [tex]\( k(x) \)[/tex] are defined as follows:
- [tex]\( h(x) = x^2 + 1 \)[/tex]
- [tex]\( k(x) = x - 2 \)[/tex]
### Step 1: Calculate [tex]\((h + k)(2)\)[/tex]
We need to find [tex]\( h(2) \)[/tex] and [tex]\( k(2) \)[/tex] and then add them together.
1. Calculate [tex]\( h(2) \)[/tex]:
[tex]\[ h(2) = (2)^2 + 1 = 4 + 1 = 5 \][/tex]
2. Calculate [tex]\( k(2) \)[/tex]:
[tex]\[ k(2) = 2 - 2 = 0 \][/tex]
3. Add [tex]\( h(2) \)[/tex] and [tex]\( k(2) \)[/tex]:
[tex]\[ (h + k)(2) = h(2) + k(2) = 5 + 0 = 5 \][/tex]
So, [tex]\((h + k)(2) = 5\)[/tex].
### Step 2: Calculate [tex]\((h - k)(3)\)[/tex]
We need to find [tex]\( h(3) \)[/tex] and [tex]\( k(3) \)[/tex] and then subtract [tex]\( k(3) \)[/tex] from [tex]\( h(3) \)[/tex].
1. Calculate [tex]\( h(3) \)[/tex]:
[tex]\[ h(3) = (3)^2 + 1 = 9 + 1 = 10 \][/tex]
2. Calculate [tex]\( k(3) \)[/tex]:
[tex]\[ k(3) = 3 - 2 = 1 \][/tex]
3. Subtract [tex]\( k(3) \)[/tex] from [tex]\( h(3) \)[/tex]:
[tex]\[ (h - k)(3) = h(3) - k(3) = 10 - 1 = 9 \][/tex]
So, [tex]\((h - k)(3) = 9\)[/tex].
### Step 3: Evaluate [tex]\( 3h(2) + 2k(3) \)[/tex]
We already have computed [tex]\( h(2) \)[/tex] and [tex]\( k(3) \)[/tex]:
1. [tex]\( h(2) = 5 \)[/tex] (from Step 1)
2. [tex]\( k(3) = 1 \)[/tex] (from Step 2)
Now calculate:
1. [tex]\( 3h(2) \)[/tex]:
[tex]\[ 3h(2) = 3 \times 5 = 15 \][/tex]
2. [tex]\( 2k(3) \)[/tex]:
[tex]\[ 2k(3) = 2 \times 1 = 2 \][/tex]
3. Add these results together:
[tex]\[ 3h(2) + 2k(3) = 15 + 2 = 17 \][/tex]
So, [tex]\( 3h(2) + 2k(3) = 17 \)[/tex].
### Summary:
- [tex]\((h + k)(2) = 5\)[/tex]
- [tex]\((h - k)(3) = 9\)[/tex]
- [tex]\( 3h(2) + 2k(3) = 17 \)[/tex]
