Sure! Let's evaluate each of the expressions step-by-step.
### Step 1: Evaluate [tex]\((h + k)(2)\)[/tex]
Given the functions:
[tex]\[ h(x) = x^2 + 1 \][/tex]
[tex]\[ k(x) = x - 2 \][/tex]
First, we need to calculate [tex]\( h(2) \)[/tex]:
[tex]\[ h(2) = 2^2 + 1 = 4 + 1 = 5 \][/tex]
Next, calculate [tex]\( k(2) \)[/tex]:
[tex]\[ k(2) = 2 - 2 = 0 \][/tex]
Now, we find [tex]\((h + k)(2)\)[/tex] by adding [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: Evaluate [tex]\((h - k)(3)\)[/tex]
First, we need to calculate [tex]\( h(3) \)[/tex]:
[tex]\[ h(3) = 3^2 + 1 = 9 + 1 = 10 \][/tex]
Next, calculate [tex]\( k(3) \)[/tex]:
[tex]\[ k(3) = 3 - 2 = 1 \][/tex]
Now, we find [tex]\((h - k)(3)\)[/tex] by subtracting [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]
First, we already have:
[tex]\[ h(2) = 5 \][/tex]
[tex]\[ k(3) = 1 \][/tex]
Now, calculate [tex]\( 3h(2) \)[/tex]:
[tex]\[ 3h(2) = 3 \times 5 = 15 \][/tex]
And calculate [tex]\( 2k(3) \)[/tex]:
[tex]\[ 2k(3) = 2 \times 1 = 2 \][/tex]
Finally, we find the value of [tex]\( 3h(2) + 2k(3) \)[/tex]:
[tex]\[ 3h(2) + 2k(3) = 15 + 2 = 17 \][/tex]
So,
[tex]\[ 3h(2) + 2k(3) = 17 \][/tex]
### Final Results
1. [tex]\((h + k)(2) = 5\)[/tex]
2. [tex]\((h - k)(3) = 9\)[/tex]
3. [tex]\(3h(2) + 2k(3) = 17\)[/tex]
Thus, the values are:
[tex]\[ (5, 9, 17) \][/tex]
