Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Get quick and reliable solutions to your questions from knowledgeable professionals on our comprehensive Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To solve this problem, we need to determine the compositions of two functions [tex]\(f(x) = x + 2\)[/tex] and [tex]\(g(x) = 3x^2 - 5x - 2\)[/tex], specifically [tex]\((f \circ g)(x)\)[/tex] and [tex]\((g \circ f)(x)\)[/tex]. Additionally, we need to establish the domain for each composition.
### Step 1: Finding [tex]\((f \circ g)(x)\)[/tex]
The composition [tex]\( (f \circ g)(x) \)[/tex] means applying [tex]\( f \)[/tex] to [tex]\( g(x) \)[/tex]. In other words, we need to substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex].
Given:
[tex]\[ g(x) = 3x^2 - 5x - 2 \][/tex]
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f(3x^2 - 5x - 2) \][/tex]
Since [tex]\( f(x) \)[/tex] is defined as:
[tex]\[ f(x) = x + 2 \][/tex]
We replace [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] with [tex]\( g(x) \)[/tex]:
[tex]\[ f(3x^2 - 5x - 2) = (3x^2 - 5x - 2) + 2 \][/tex]
Simplify the expression:
[tex]\[ f(g(x)) = 3x^2 - 5x \][/tex]
Therefore:
[tex]\[ (f \circ g)(x) = 3x^2 - 5x \][/tex]
### Step 2: Finding [tex]\((g \circ f)(x)\)[/tex]
The composition [tex]\( (g \circ f)(x) \)[/tex] means applying [tex]\( g \)[/tex] to [tex]\( f(x) \)[/tex]. This time, we need to substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex].
Given:
[tex]\[ f(x) = x + 2 \][/tex]
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g(x + 2) \][/tex]
Since [tex]\( g(x) \)[/tex] is defined as:
[tex]\[ g(x) = 3x^2 - 5x - 2 \][/tex]
We replace [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex] with [tex]\( f(x) \)[/tex]:
[tex]\[ g(x + 2) = 3(x + 2)^2 - 5(x + 2) - 2 \][/tex]
Expand and simplify the quadratic expression:
[tex]\[ g(x + 2) = 3(x^2 + 4x + 4) - 5(x + 2) - 2 \][/tex]
[tex]\[ = 3x^2 + 12x + 12 - 5x - 10 - 2 \][/tex]
[tex]\[ = 3x^2 + 7x \][/tex]
Therefore:
[tex]\[ (g \circ f)(x) = 3x^2 + 7x \][/tex]
### Step 3: Determining the Domains
Both [tex]\( f(x) = x + 2 \)[/tex] and [tex]\( g(x) = 3x^2 - 5x - 2 \)[/tex] are polynomial functions. Polynomial functions are defined for all real numbers.
Thus, the domain of both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is the set of all real numbers, [tex]\(\mathbb{R}\)[/tex].
Since both [tex]\( (f \circ g)(x) \)[/tex] and [tex]\( (g \circ f)(x) \)[/tex] are compositions of these polynomial functions, their domains are also all real numbers.
Therefore, the domains are:
[tex]\[ \text{Domain of } (f \circ g)(x) = \mathbb{R} \][/tex]
[tex]\[ \text{Domain of } (g \circ f)(x) = \mathbb{R} \][/tex]
### Summary:
1. [tex]\( (f \circ g)(x) = 3x^2 - 5x \)[/tex]
2. [tex]\( (g \circ f)(x) = 3x^2 + 7x \)[/tex]
3. The domain of [tex]\( (f \circ g)(x) = \mathbb{R} \)[/tex]
4. The domain of [tex]\( (g \circ f)(x) = \mathbb{R} \)[/tex]
These are the compositions and their respective domains.
### Step 1: Finding [tex]\((f \circ g)(x)\)[/tex]
The composition [tex]\( (f \circ g)(x) \)[/tex] means applying [tex]\( f \)[/tex] to [tex]\( g(x) \)[/tex]. In other words, we need to substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex].
Given:
[tex]\[ g(x) = 3x^2 - 5x - 2 \][/tex]
Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f(3x^2 - 5x - 2) \][/tex]
Since [tex]\( f(x) \)[/tex] is defined as:
[tex]\[ f(x) = x + 2 \][/tex]
We replace [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] with [tex]\( g(x) \)[/tex]:
[tex]\[ f(3x^2 - 5x - 2) = (3x^2 - 5x - 2) + 2 \][/tex]
Simplify the expression:
[tex]\[ f(g(x)) = 3x^2 - 5x \][/tex]
Therefore:
[tex]\[ (f \circ g)(x) = 3x^2 - 5x \][/tex]
### Step 2: Finding [tex]\((g \circ f)(x)\)[/tex]
The composition [tex]\( (g \circ f)(x) \)[/tex] means applying [tex]\( g \)[/tex] to [tex]\( f(x) \)[/tex]. This time, we need to substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex].
Given:
[tex]\[ f(x) = x + 2 \][/tex]
Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g(x + 2) \][/tex]
Since [tex]\( g(x) \)[/tex] is defined as:
[tex]\[ g(x) = 3x^2 - 5x - 2 \][/tex]
We replace [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex] with [tex]\( f(x) \)[/tex]:
[tex]\[ g(x + 2) = 3(x + 2)^2 - 5(x + 2) - 2 \][/tex]
Expand and simplify the quadratic expression:
[tex]\[ g(x + 2) = 3(x^2 + 4x + 4) - 5(x + 2) - 2 \][/tex]
[tex]\[ = 3x^2 + 12x + 12 - 5x - 10 - 2 \][/tex]
[tex]\[ = 3x^2 + 7x \][/tex]
Therefore:
[tex]\[ (g \circ f)(x) = 3x^2 + 7x \][/tex]
### Step 3: Determining the Domains
Both [tex]\( f(x) = x + 2 \)[/tex] and [tex]\( g(x) = 3x^2 - 5x - 2 \)[/tex] are polynomial functions. Polynomial functions are defined for all real numbers.
Thus, the domain of both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is the set of all real numbers, [tex]\(\mathbb{R}\)[/tex].
Since both [tex]\( (f \circ g)(x) \)[/tex] and [tex]\( (g \circ f)(x) \)[/tex] are compositions of these polynomial functions, their domains are also all real numbers.
Therefore, the domains are:
[tex]\[ \text{Domain of } (f \circ g)(x) = \mathbb{R} \][/tex]
[tex]\[ \text{Domain of } (g \circ f)(x) = \mathbb{R} \][/tex]
### Summary:
1. [tex]\( (f \circ g)(x) = 3x^2 - 5x \)[/tex]
2. [tex]\( (g \circ f)(x) = 3x^2 + 7x \)[/tex]
3. The domain of [tex]\( (f \circ g)(x) = \mathbb{R} \)[/tex]
4. The domain of [tex]\( (g \circ f)(x) = \mathbb{R} \)[/tex]
These are the compositions and their respective domains.
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
