Certainly! Let's solve this step-by-step.
First, we have the functions:
[tex]\[ h(x) = x^2 + 1 \][/tex]
[tex]\[ k(x) = x - 2 \][/tex]
### Step 1: Calculate [tex]\((h + k)(2)\)[/tex]
First, we 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]
Now, sum these results:
[tex]\[ (h + k)(2) = h(2) + k(2) = 5 + 0 = 5 \][/tex]
### Step 2: Calculate [tex]\((h - k)(3)\)[/tex]
Next, we 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]
Now, subtract these results:
[tex]\[ (h - k)(3) = h(3) - k(3) = 10 - 1 = 9 \][/tex]
### Step 3: Evaluate [tex]\( 3h(2) + 2k(3) \)[/tex]
We already know from the first step that:
[tex]\[ h(2) = 5 \][/tex]
And from the second step, we know that:
[tex]\[ k(3) = 1 \][/tex]
Now, multiply and sum these values:
[tex]\[ 3h(2) = 3 \times 5 = 15 \][/tex]
[tex]\[ 2k(3) = 2 \times 1 = 2 \][/tex]
Add the results:
[tex]\[ 3h(2) + 2k(3) = 15 + 2 = 17 \][/tex]
So, the final results are:
[tex]\[ (h + k)(2) = 5 \][/tex]
[tex]\[ (h - k)(3) = 9 \][/tex]
[tex]\[ 3h(2) + 2k(3) = 17 \][/tex]
