Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
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].
### 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].
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.