Alright, let's solve each of these problems step-by-step to find [tex]\((f-g)(x)\)[/tex] for the given functions. We will use [tex]\(x = 2\)[/tex] as the value to calculate.
### Problem 1
Given:
[tex]\[ f(x) = 3x + 3 \][/tex]
[tex]\[ g(x) = -4x + 1 \][/tex]
We need to find [tex]\( (f - g)(x) \)[/tex].
Step 1: Calculate [tex]\( f(2) \)[/tex].
[tex]\[ f(2) = 3(2) + 3 = 6 + 3 = 9 \][/tex]
Step 2: Calculate [tex]\( g(2) \)[/tex].
[tex]\[ g(2) = -4(2) + 1 = -8 + 1 = -7 \][/tex]
Step 3: Find [tex]\( (f - g)(2) \)[/tex], which is [tex]\( f(2) - g(2) \)[/tex].
[tex]\[ (f - g)(2) = 9 - (-7) = 9 + 7 = 16 \][/tex]
Thus, the result for Problem 1 is:
[tex]\[ (f - g)(2) = 16 \][/tex]
### Problem 2
Given:
[tex]\[ f(x) = 2x + 5 \][/tex]
[tex]\[ g(x) = 4x^2 + 2x - 2 \][/tex]
We need to find [tex]\( (f - g)(x) \)[/tex].
Step 1: Calculate [tex]\( f(2) \)[/tex].
[tex]\[ f(2) = 2(2) + 5 = 4 + 5 = 9 \][/tex]
Step 2: Calculate [tex]\( g(2) \)[/tex].
[tex]\[ g(2) = 4(2)^2 + 2(2) - 2 = 4(4) + 4 - 2 = 16 + 4 - 2 = 18 \][/tex]
Step 3: Find [tex]\( (f - g)(2) \)[/tex], which is [tex]\( f(2) - g(2) \)[/tex].
[tex]\[ (f - g)(2) = 9 - 18 = -9 \][/tex]
Thus, the result for Problem 2 is:
[tex]\[ (f - g)(2) = -9 \][/tex]
### Problem 3
Given:
[tex]\[ f(x) = -15x^3 - 2x + 5 \][/tex]
[tex]\[ g(x) = 3x^2 + x - 7 \][/tex]
We need to find [tex]\( (f - g)(x) \)[/tex].
Step 1: Calculate [tex]\( f(2) \)[/tex].
[tex]\[ f(2) = -15(2)^3 - 2(2) + 5 = -15(8) - 4 + 5 = -120 - 4 + 5 = -119 \][/tex]
Step 2: Calculate [tex]\( g(2) \)[/tex].
[tex]\[ g(2) = 3(2)^2 + 2 - 7 = 3(4) + 2 - 7 = 12 + 2 - 7 = 7 \][/tex]
Step 3: Find [tex]\( (f - g)(2) \)[/tex], which is [tex]\( f(2) - g(2) \)[/tex].
[tex]\[ (f - g)(2) = -119 - 7 = -126 \][/tex]
Thus, the result for Problem 3 is:
[tex]\[ (f - g)(2) = -126 \][/tex]
### Summary of Results
- For Problem 1: [tex]\( (f - g)(2) = 16 \)[/tex]
- For Problem 2: [tex]\( (f - g)(2) = -9 \)[/tex]
- For Problem 3: [tex]\( (f - g)(2) = -126 \)[/tex]
These are the correct values for [tex]\((f - g)(x)\)[/tex] for each given set of functions evaluated at [tex]\(x = 2\)[/tex].
