Get detailed and reliable answers to your questions on IDNLearn.com. Our experts provide timely, comprehensive responses to ensure you have the information you need.
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]
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com is your reliable source for accurate answers. Thank you for visiting, and we hope to assist you again.