Let's solve the given problems step-by-step using algebraic methods.
### Problem 1:
We have two functions:
[tex]\[ f(x) = 2x + 3 \][/tex]
[tex]\[ g(x) = 3x - 2 \][/tex]
We need to find [tex]\((f + g)(2)\)[/tex] and [tex]\((f - g)(-2)\)[/tex].
#### (f + g)(2):
First, evaluate [tex]\(f(2)\)[/tex]:
[tex]\[ f(2) = 2(2) + 3 = 4 + 3 = 7 \][/tex]
Next, evaluate [tex]\(g(2)\)[/tex]:
[tex]\[ g(2) = 3(2) - 2 = 6 - 2 = 4 \][/tex]
Now, add the results:
[tex]\[ (f + g)(2) = f(2) + g(2) = 7 + 4 = 11 \][/tex]
#### (f - g)(-2):
First, evaluate [tex]\(f(-2)\)[/tex]:
[tex]\[ f(-2) = 2(-2) + 3 = -4 + 3 = -1 \][/tex]
Next, evaluate [tex]\(g(-2)\)[/tex]:
[tex]\[ g(-2) = 3(-2) - 2 = -6 - 2 = -8 \][/tex]
Now, subtract the results:
[tex]\[ (f - g)(-2) = f(-2) - g(-2) = -1 - (-8) = -1 + 8 = 7 \][/tex]
### Problem 2:
We have two functions:
[tex]\[ f(x) = x^2 - 3x + 2 \][/tex]
[tex]\[ g(x) = 2x^2 - x + 1 \][/tex]
We need to find [tex]\((f + g)(2)\)[/tex] and [tex]\((f - g)(-2)\)[/tex].
#### (f + g)(2):
First, evaluate [tex]\(f(2)\)[/tex]:
[tex]\[ f(2) = 2^2 - 3(2) + 2 = 4 - 6 + 2 = 0 \][/tex]
Next, evaluate [tex]\(g(2)\)[/tex]:
[tex]\[ g(2) = 2(2^2) - 2 + 1 = 2 \cdot 4 - 2 + 1 = 8 - 2 + 1 = 7 \][/tex]
Now, add the results:
[tex]\[ (f + g)(2) = f(2) + g(2) = 0 + 7 = 7 \][/tex]
#### (f - g)(-2):
First, evaluate [tex]\(f(-2)\)[/tex]:
[tex]\[ f(-2) = (-2)^2 - 3(-2) + 2 = 4 + 6 + 2 = 12 \][/tex]
Next, evaluate [tex]\(g(-2)\)[/tex]:
[tex]\[ g(-2) = 2(-2)^2 - (-2) + 1 = 2 \cdot 4 + 2 + 1 = 8 + 2 + 1 = 11 \][/tex]
Now, subtract the results:
[tex]\[ (f - g)(-2) = f(-2) - g(-2) = 12 - 11 = 1 \][/tex]
### Summary:
- For [tex]\(f(x) = 2x + 3\)[/tex] and [tex]\(g(x) = 3x - 2\)[/tex]:
[tex]\[
(f + g)(2) = 11, \quad (f - g)(-2) = 7
\][/tex]
- For [tex]\(f(x) = x^2 - 3x + 2\)[/tex] and [tex]\(g(x) = 2x^2 - x + 1\)[/tex]:
[tex]\[
(f + g)(2) = 7, \quad (f - g)(-2) = 1
\][/tex]
