Let's solve the following problems step-by-step.
### Problem 1:
We are given the functions [tex]\( f(x) = x^2 + 4 \)[/tex] and [tex]\( g(x) = x - 1 \)[/tex]. We need to find [tex]\( (g+f)(x) \)[/tex].
To find [tex]\( (g+f)(x) \)[/tex], we need to add [tex]\( g(x) \)[/tex] and [tex]\( f(x) \)[/tex]:
[tex]\[ (g+f)(x) = g(x) + f(x) \][/tex]
Substitute the given functions:
[tex]\[ g(x) = x - 1 \][/tex]
[tex]\[ f(x) = x^2 + 4 \][/tex]
Thus,
[tex]\[ (g+f)(x) = (x - 1) + (x^2 + 4) \][/tex]
[tex]\[ (g+f)(x) = x^2 + x + 3 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ (g+f)(2) = 2^2 + 2 + 3 \][/tex]
[tex]\[ (g+f)(2) = 4 + 2 + 3 \][/tex]
[tex]\[ (g+f)(2) = 9 \][/tex]
So, the value is [tex]\( 9 \)[/tex].
### Problem 2:
We are given the functions [tex]\( g(x) = 4x - 1 \)[/tex] and [tex]\( f(x) = x - 2 \)[/tex]. We need to find [tex]\( g(x) - f(x) \)[/tex].
To find [tex]\( g(x) - f(x) \)[/tex], we simply subtract [tex]\( f(x) \)[/tex] from [tex]\( g(x) \)[/tex]:
[tex]\[ (g-f)(x) = g(x) - f(x) \][/tex]
Substitute the given functions:
[tex]\[ g(x) = 4x - 1 \][/tex]
[tex]\[ f(x) = x - 2 \][/tex]
Thus,
[tex]\[ (g-f)(x) = (4x - 1) - (x - 2) \][/tex]
[tex]\[ (g-f)(x) = 4x - 1 - x + 2 \][/tex]
[tex]\[ (g-f)(x) = 3x + 1 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ (g-f)(2) = 3 \cdot 2 + 1 \][/tex]
[tex]\[ (g-f)(2) = 6 + 1 \][/tex]
[tex]\[ (g-f)(2) = 7 \][/tex]
So, the value is [tex]\( 7 \)[/tex].
### Problem 3:
We are given the functions [tex]\( g(x) = 3x + 1 \)[/tex] and [tex]\( h(x) = 2x - 2 \)[/tex]. We need to find [tex]\( (g-h)(x) \)[/tex].
To find [tex]\( (g-h)(x) \)[/tex], we need to subtract [tex]\( h(x) \)[/tex] from [tex]\( g(x) \)[/tex]:
[tex]\[ (g-h)(x) = g(x) - h(x) \][/tex]
Substitute the given functions:
[tex]\[ g(x) = 3x + 1 \][/tex]
[tex]\[ h(x) = 2x - 2 \][/tex]
Thus,
[tex]\[ (g-h)(x) = (3x + 1) - (2x - 2) \][/tex]
[tex]\[ (g-h)(x) = 3x + 1 - 2x + 2 \][/tex]
[tex]\[ (g-h)(x) = x + 3 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ (g-h)(2) = 2 + 3 \][/tex]
[tex]\[ (g-h)(2) = 5 \][/tex]
So, the value is [tex]\( 5 \)[/tex].
### Problem 4:
We are given the functions [tex]\( g(x) = x^2 - 2x - 1 \)[/tex] and [tex]\( h(x) = 2x - 2 \)[/tex]. We need to find [tex]\( g(x) + h(x) \)[/tex].
To find [tex]\( g(x) + h(x) \)[/tex], we need to add [tex]\( g(x) \)[/tex] and [tex]\( h(x) \)[/tex]:
[tex]\[ (g+h)(x) = g(x) + h(x) \][/tex]
Substitute the given functions:
[tex]\[ g(x) = x^2 - 2x - 1 \][/tex]
[tex]\[ h(x) = 2x - 2 \][/tex]
Thus,
[tex]\[ (g+h)(x) = (x^2 - 2x - 1) + (2x - 2) \][/tex]
[tex]\[ (g+h)(x) = x^2 - 2x - 1 + 2x - 2 \][/tex]
[tex]\[ (g+h)(x) = x^2 - 3 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ (g+h)(2) = 2^2 - 3 \][/tex]
[tex]\[ (g+h)(2) = 4 - 3 \][/tex]
[tex]\[ (g+h)(2) = 1 \][/tex]
So, the value is [tex]\( 1 \)[/tex].
