Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Get detailed answers to your questions from a community of experts dedicated to providing accurate information. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
Let's analyze and calculate each part step-by-step using the functions [tex]\( f(x) = x - 3 \)[/tex] and [tex]\( g(x) = 5x^2 - 2 \)[/tex].
### Part (a): [tex]\((f \circ g)(x)\)[/tex]
The composition of two functions [tex]\( (f \circ g)(x) \)[/tex] means we first apply [tex]\( g(x) \)[/tex] and then apply [tex]\( f \)[/tex] to the result. In other words, [tex]\( (f \circ g)(x) = f(g(x)) \)[/tex].
Given:
[tex]\[ g(x) = 5x^2 - 2 \][/tex]
Now, apply [tex]\( f \)[/tex] to [tex]\( g(x) \)[/tex]:
[tex]\[ f(g(x)) = f(5x^2 - 2) \][/tex]
Since [tex]\( f(x) = x - 3 \)[/tex], replace [tex]\( x \)[/tex] in [tex]\( f \)[/tex] with [tex]\( 5x^2 - 2 \)[/tex]:
[tex]\[ f(5x^2 - 2) = (5x^2 - 2) - 3 \][/tex]
[tex]\[ f(5x^2 - 2) = 5x^2 - 5 \][/tex]
So:
[tex]\[ (f \circ g)(x) = 5x^2 - 5 \][/tex]
### Part (b): [tex]\((g \circ f)(x)\)[/tex]
The composition of two functions [tex]\( (g \circ f)(x) \)[/tex] means we first apply [tex]\( f(x) \)[/tex] and then apply [tex]\( g \)[/tex] to the result. In other words, [tex]\( (g \circ f)(x) = g(f(x)) \)[/tex].
Given:
[tex]\[ f(x) = x - 3 \][/tex]
Now, apply [tex]\( g \)[/tex] to [tex]\( f(x) \)[/tex]:
[tex]\[ g(f(x)) = g(x - 3) \][/tex]
Since [tex]\( g(x) = 5x^2 - 2 \)[/tex], replace [tex]\( x \)[/tex] in [tex]\( g \)[/tex] with [tex]\( x - 3 \)[/tex]:
[tex]\[ g(x - 3) = 5(x - 3)^2 - 2 \][/tex]
[tex]\[ g(x - 3) = 5(x^2 - 6x + 9) - 2 \][/tex]
[tex]\[ g(x - 3) = 5x^2 - 30x + 45 - 2 \][/tex]
[tex]\[ g(x - 3) = 5x^2 - 30x + 43 \][/tex]
So:
[tex]\[ (g \circ f)(x) = 5x^2 - 30x + 43 \][/tex]
### Part (c): [tex]\((f \circ g)(1)\)[/tex]
To find [tex]\((f \circ g)(1)\)[/tex], we first find [tex]\( g(1) \)[/tex] and then apply [tex]\( f \)[/tex] to the result.
First, calculate [tex]\( g(1) \)[/tex]:
[tex]\[ g(1) = 5 \cdot 1^2 - 2 \][/tex]
[tex]\[ g(1) = 5 - 2 \][/tex]
[tex]\[ g(1) = 3 \][/tex]
Now, apply [tex]\( f \)[/tex] to the result:
[tex]\[ f(g(1)) = f(3) \][/tex]
[tex]\[ f(3) = 3 - 3 \][/tex]
[tex]\[ f(3) = 0 \][/tex]
So:
[tex]\[ (f \circ g)(1) = 0 \][/tex]
### Part (d): [tex]\((g \circ f)(1)\)[/tex]
To find [tex]\((g \circ f)(1)\)[/tex], we first find [tex]\( f(1) \)[/tex] and then apply [tex]\( g \)[/tex] to the result.
First, calculate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 1 - 3 \][/tex]
[tex]\[ f(1) = -2 \][/tex]
Now, apply [tex]\( g \)[/tex] to the result:
[tex]\[ g(f(1)) = g(-2) \][/tex]
[tex]\[ g(-2) = 5 \cdot (-2)^2 - 2 \][/tex]
[tex]\[ g(-2) = 5 \cdot 4 - 2 \][/tex]
[tex]\[ g(-2) = 20 - 2 \][/tex]
[tex]\[ g(-2) = 18 \][/tex]
So:
[tex]\[ (g \circ f)(1) = 18 \][/tex]
### Summary
The functions and their evaluations are as follows:
a. [tex]\((f \circ g)(x) = 5x^2 - 5\)[/tex]
b. [tex]\((g \circ f)(x) = 5x^2 - 30x + 43\)[/tex]
c. [tex]\((f \circ g)(1) = 0\)[/tex]
d. [tex]\((g \circ f)(1) = 18\)[/tex]
### Part (a): [tex]\((f \circ g)(x)\)[/tex]
The composition of two functions [tex]\( (f \circ g)(x) \)[/tex] means we first apply [tex]\( g(x) \)[/tex] and then apply [tex]\( f \)[/tex] to the result. In other words, [tex]\( (f \circ g)(x) = f(g(x)) \)[/tex].
Given:
[tex]\[ g(x) = 5x^2 - 2 \][/tex]
Now, apply [tex]\( f \)[/tex] to [tex]\( g(x) \)[/tex]:
[tex]\[ f(g(x)) = f(5x^2 - 2) \][/tex]
Since [tex]\( f(x) = x - 3 \)[/tex], replace [tex]\( x \)[/tex] in [tex]\( f \)[/tex] with [tex]\( 5x^2 - 2 \)[/tex]:
[tex]\[ f(5x^2 - 2) = (5x^2 - 2) - 3 \][/tex]
[tex]\[ f(5x^2 - 2) = 5x^2 - 5 \][/tex]
So:
[tex]\[ (f \circ g)(x) = 5x^2 - 5 \][/tex]
### Part (b): [tex]\((g \circ f)(x)\)[/tex]
The composition of two functions [tex]\( (g \circ f)(x) \)[/tex] means we first apply [tex]\( f(x) \)[/tex] and then apply [tex]\( g \)[/tex] to the result. In other words, [tex]\( (g \circ f)(x) = g(f(x)) \)[/tex].
Given:
[tex]\[ f(x) = x - 3 \][/tex]
Now, apply [tex]\( g \)[/tex] to [tex]\( f(x) \)[/tex]:
[tex]\[ g(f(x)) = g(x - 3) \][/tex]
Since [tex]\( g(x) = 5x^2 - 2 \)[/tex], replace [tex]\( x \)[/tex] in [tex]\( g \)[/tex] with [tex]\( x - 3 \)[/tex]:
[tex]\[ g(x - 3) = 5(x - 3)^2 - 2 \][/tex]
[tex]\[ g(x - 3) = 5(x^2 - 6x + 9) - 2 \][/tex]
[tex]\[ g(x - 3) = 5x^2 - 30x + 45 - 2 \][/tex]
[tex]\[ g(x - 3) = 5x^2 - 30x + 43 \][/tex]
So:
[tex]\[ (g \circ f)(x) = 5x^2 - 30x + 43 \][/tex]
### Part (c): [tex]\((f \circ g)(1)\)[/tex]
To find [tex]\((f \circ g)(1)\)[/tex], we first find [tex]\( g(1) \)[/tex] and then apply [tex]\( f \)[/tex] to the result.
First, calculate [tex]\( g(1) \)[/tex]:
[tex]\[ g(1) = 5 \cdot 1^2 - 2 \][/tex]
[tex]\[ g(1) = 5 - 2 \][/tex]
[tex]\[ g(1) = 3 \][/tex]
Now, apply [tex]\( f \)[/tex] to the result:
[tex]\[ f(g(1)) = f(3) \][/tex]
[tex]\[ f(3) = 3 - 3 \][/tex]
[tex]\[ f(3) = 0 \][/tex]
So:
[tex]\[ (f \circ g)(1) = 0 \][/tex]
### Part (d): [tex]\((g \circ f)(1)\)[/tex]
To find [tex]\((g \circ f)(1)\)[/tex], we first find [tex]\( f(1) \)[/tex] and then apply [tex]\( g \)[/tex] to the result.
First, calculate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = 1 - 3 \][/tex]
[tex]\[ f(1) = -2 \][/tex]
Now, apply [tex]\( g \)[/tex] to the result:
[tex]\[ g(f(1)) = g(-2) \][/tex]
[tex]\[ g(-2) = 5 \cdot (-2)^2 - 2 \][/tex]
[tex]\[ g(-2) = 5 \cdot 4 - 2 \][/tex]
[tex]\[ g(-2) = 20 - 2 \][/tex]
[tex]\[ g(-2) = 18 \][/tex]
So:
[tex]\[ (g \circ f)(1) = 18 \][/tex]
### Summary
The functions and their evaluations are as follows:
a. [tex]\((f \circ g)(x) = 5x^2 - 5\)[/tex]
b. [tex]\((g \circ f)(x) = 5x^2 - 30x + 43\)[/tex]
c. [tex]\((f \circ g)(1) = 0\)[/tex]
d. [tex]\((g \circ f)(1) = 18\)[/tex]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
