Let's break down the given functions and expressions step by step.
First, we're given two functions:
[tex]\[ h(x) = x^2 + 1 \][/tex]
[tex]\[ k(x) = x - 2 \][/tex]
### Step 1: Calculate [tex]\((h + k)(2)\)[/tex]
To find [tex]\((h + k)(2)\)[/tex], we need to evaluate both [tex]\(h(2)\)[/tex] and [tex]\(k(2)\)[/tex] first, and then add them together.
1. Evaluate [tex]\(h(2)\)[/tex]:
[tex]\[ h(2) = 2^2 + 1 = 4 + 1 = 5 \][/tex]
2. Evaluate [tex]\(k(2)\)[/tex]:
[tex]\[ k(2) = 2 - 2 = 0 \][/tex]
Add these two results:
[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]
To find [tex]\((h - k)(3)\)[/tex], we need to evaluate both [tex]\(h(3)\)[/tex] and [tex]\(k(3)\)[/tex] first, and then subtract [tex]\(k(3)\)[/tex] from [tex]\(h(3)\)[/tex].
1. Evaluate [tex]\(h(3)\)[/tex]:
[tex]\[ h(3) = 3^2 + 1 = 9 + 1 = 10 \][/tex]
2. Evaluate [tex]\(k(3)\)[/tex]:
[tex]\[ k(3) = 3 - 2 = 1 \][/tex]
Subtract these results:
[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]
Now, let's calculate [tex]\(3h(2) + 2k(3)\)[/tex].
1. Evaluate [tex]\(h(2)\)[/tex] and [tex]\(k(3)\)[/tex] which we previously found:
[tex]\[ h(2) = 5 \][/tex]
[tex]\[ k(3) = 1 \][/tex]
2. Substitute these values into the expression:
[tex]\[ 3h(2) + 2k(3) = 3 \cdot 5 + 2 \cdot 1 = 15 + 2 = 17 \][/tex]
So, [tex]\(3h(2) + 2k(3) = 17\)[/tex].
Putting all these together:
- [tex]\((h + k)(2) = 5\)[/tex]
- [tex]\((h - k)(3) = 9\)[/tex]
- [tex]\(3h(2) + 2k(3) = 17\)[/tex]
These are the detailed computations for the given expressions.
