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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.
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