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Sagot :
Let's solve the problem step-by-step.
### Given Functions:
[tex]\[ f(x) = 4x^3 \][/tex]
[tex]\[ g(x) = x^2 + 4x - 3 \][/tex]
### Part (a): [tex]\((f \circ g)(x)\)[/tex]
The notation [tex]\((f \circ g)(x)\)[/tex] means we want to find [tex]\(f(g(x))\)[/tex].
1. First, we express [tex]\(g(x)\)[/tex]:
[tex]\[ g(x) = x^2 + 4x - 3 \][/tex]
2. Next, we substitute [tex]\(g(x)\)[/tex] into the function [tex]\(f(x)\)[/tex]:
[tex]\[ f(g(x)) = f(x^2 + 4x - 3) \][/tex]
3. Now, use the definition of [tex]\(f\)[/tex]:
[tex]\[ f(x) = 4x^3 \][/tex]
So we need to find [tex]\(f(x^2 + 4x - 3)\)[/tex].
4. Replace all instances of [tex]\(x\)[/tex] in [tex]\(f(x) = 4x^3\)[/tex] with [tex]\(x^2 + 4x - 3\)[/tex]:
[tex]\[ f(x^2 + 4x - 3) = 4(x^2 + 4x - 3)^3 \][/tex]
So the solution for [tex]\((f \circ g)(x)\)[/tex] is:
[tex]\[ (f \circ g)(x) = 4(x^2 + 4x - 3)^3 \][/tex]
### Part (b): [tex]\((g \circ f)(x)\)[/tex]
The notation [tex]\((g \circ f)(x)\)[/tex] means we want to find [tex]\(g(f(x))\)[/tex].
1. First, we express [tex]\(f(x)\)[/tex]:
[tex]\[ f(x) = 4x^3 \][/tex]
2. Next, we substitute [tex]\(f(x)\)[/tex] into the function [tex]\(g(x)\)[/tex]:
[tex]\[ g(f(x)) = g(4x^3) \][/tex]
3. Now, use the definition of [tex]\(g\)[/tex]:
[tex]\[ g(x) = x^2 + 4x - 3 \][/tex]
So we need to find [tex]\(g(4x^3)\)[/tex].
4. Replace all instances of [tex]\(x\)[/tex] in [tex]\(g(x) = x^2 + 4x - 3\)[/tex] with [tex]\(4x^3\)[/tex]:
[tex]\[ g(4x^3) = (4x^3)^2 + 4(4x^3) - 3 \][/tex]
[tex]\[ g(4x^3) = 16x^6 + 16x^3 - 3 \][/tex]
So the solution for [tex]\((g \circ f)(x)\)[/tex] is:
[tex]\[ (g \circ f)(x) = 16x^6 + 16x^3 - 3 \][/tex]
### Part (c): [tex]\((f \circ f)(x)\)[/tex]
The notation [tex]\((f \circ f)(x)\)[/tex] means we want to find [tex]\(f(f(x))\)[/tex].
1. First, we express [tex]\(f(x)\)[/tex]:
[tex]\[ f(x) = 4x^3 \][/tex]
2. Next, we substitute [tex]\(f(x)\)[/tex] into the function [tex]\(f(x)\)[/tex]:
[tex]\[ f(f(x)) = f(4x^3) \][/tex]
3. Now, use the definition of [tex]\(f\)[/tex]:
[tex]\[ f(x) = 4x^3 \][/tex]
So we need to find [tex]\(f(4x^3)\)[/tex].
4. Replace all instances of [tex]\(x\)[/tex] in [tex]\(f(x) = 4x^3\)[/tex] with [tex]\(4x^3\)[/tex]:
[tex]\[ f(4x^3) = 4(4x^3)^3 \][/tex]
[tex]\[ f(4x^3) = 4(64x^9) = 256x^9 \][/tex]
So the solution for [tex]\((f \circ f)(x)\)[/tex] is:
[tex]\[ (f \circ f)(x) = 256x^9 \][/tex]
### Domain of Each Composite Function:
1. [tex]\( (f \circ g)(x) \)[/tex]:
The domain of [tex]\(g(x)\)[/tex] is all real numbers because it is a polynomial. The output of [tex]\(g(x)\)[/tex] must also be within the domain of [tex]\(f(x)\)[/tex], which is all real numbers for [tex]\(4x^3\)[/tex]. Thus, the domain of [tex]\((f \circ g)(x)\)[/tex] is all real numbers.
2. [tex]\( (g \circ f)(x) \)[/tex]:
The domain of [tex]\(f(x)\)[/tex] is all real numbers. The output of [tex]\(f(x)\)[/tex] must be within the domain of [tex]\(g(x)\)[/tex], which is all real numbers for [tex]\(x^2 + 4x - 3\)[/tex]. Thus, the domain of [tex]\((g \circ f)(x)\)[/tex] is all real numbers.
3. [tex]\( (f \circ f)(x) \)[/tex]:
The domain of [tex]\(f(x)\)[/tex] is all real numbers. The output of [tex]\(f(x)\)[/tex] must be within the domain of itself, which is all real numbers for [tex]\(4x^3\)[/tex]. Thus, the domain of [tex]\((f \circ f)(x)\)[/tex] is all real numbers.
So, the domains for [tex]\((f \circ g)(x)\)[/tex], [tex]\((g \circ f)(x)\)[/tex], and [tex]\((f \circ f)(x)\)[/tex] are all real numbers, which can be written as:
[tex]\[ \text{Domain: } (-\infty, \infty) \][/tex]
### Given Functions:
[tex]\[ f(x) = 4x^3 \][/tex]
[tex]\[ g(x) = x^2 + 4x - 3 \][/tex]
### Part (a): [tex]\((f \circ g)(x)\)[/tex]
The notation [tex]\((f \circ g)(x)\)[/tex] means we want to find [tex]\(f(g(x))\)[/tex].
1. First, we express [tex]\(g(x)\)[/tex]:
[tex]\[ g(x) = x^2 + 4x - 3 \][/tex]
2. Next, we substitute [tex]\(g(x)\)[/tex] into the function [tex]\(f(x)\)[/tex]:
[tex]\[ f(g(x)) = f(x^2 + 4x - 3) \][/tex]
3. Now, use the definition of [tex]\(f\)[/tex]:
[tex]\[ f(x) = 4x^3 \][/tex]
So we need to find [tex]\(f(x^2 + 4x - 3)\)[/tex].
4. Replace all instances of [tex]\(x\)[/tex] in [tex]\(f(x) = 4x^3\)[/tex] with [tex]\(x^2 + 4x - 3\)[/tex]:
[tex]\[ f(x^2 + 4x - 3) = 4(x^2 + 4x - 3)^3 \][/tex]
So the solution for [tex]\((f \circ g)(x)\)[/tex] is:
[tex]\[ (f \circ g)(x) = 4(x^2 + 4x - 3)^3 \][/tex]
### Part (b): [tex]\((g \circ f)(x)\)[/tex]
The notation [tex]\((g \circ f)(x)\)[/tex] means we want to find [tex]\(g(f(x))\)[/tex].
1. First, we express [tex]\(f(x)\)[/tex]:
[tex]\[ f(x) = 4x^3 \][/tex]
2. Next, we substitute [tex]\(f(x)\)[/tex] into the function [tex]\(g(x)\)[/tex]:
[tex]\[ g(f(x)) = g(4x^3) \][/tex]
3. Now, use the definition of [tex]\(g\)[/tex]:
[tex]\[ g(x) = x^2 + 4x - 3 \][/tex]
So we need to find [tex]\(g(4x^3)\)[/tex].
4. Replace all instances of [tex]\(x\)[/tex] in [tex]\(g(x) = x^2 + 4x - 3\)[/tex] with [tex]\(4x^3\)[/tex]:
[tex]\[ g(4x^3) = (4x^3)^2 + 4(4x^3) - 3 \][/tex]
[tex]\[ g(4x^3) = 16x^6 + 16x^3 - 3 \][/tex]
So the solution for [tex]\((g \circ f)(x)\)[/tex] is:
[tex]\[ (g \circ f)(x) = 16x^6 + 16x^3 - 3 \][/tex]
### Part (c): [tex]\((f \circ f)(x)\)[/tex]
The notation [tex]\((f \circ f)(x)\)[/tex] means we want to find [tex]\(f(f(x))\)[/tex].
1. First, we express [tex]\(f(x)\)[/tex]:
[tex]\[ f(x) = 4x^3 \][/tex]
2. Next, we substitute [tex]\(f(x)\)[/tex] into the function [tex]\(f(x)\)[/tex]:
[tex]\[ f(f(x)) = f(4x^3) \][/tex]
3. Now, use the definition of [tex]\(f\)[/tex]:
[tex]\[ f(x) = 4x^3 \][/tex]
So we need to find [tex]\(f(4x^3)\)[/tex].
4. Replace all instances of [tex]\(x\)[/tex] in [tex]\(f(x) = 4x^3\)[/tex] with [tex]\(4x^3\)[/tex]:
[tex]\[ f(4x^3) = 4(4x^3)^3 \][/tex]
[tex]\[ f(4x^3) = 4(64x^9) = 256x^9 \][/tex]
So the solution for [tex]\((f \circ f)(x)\)[/tex] is:
[tex]\[ (f \circ f)(x) = 256x^9 \][/tex]
### Domain of Each Composite Function:
1. [tex]\( (f \circ g)(x) \)[/tex]:
The domain of [tex]\(g(x)\)[/tex] is all real numbers because it is a polynomial. The output of [tex]\(g(x)\)[/tex] must also be within the domain of [tex]\(f(x)\)[/tex], which is all real numbers for [tex]\(4x^3\)[/tex]. Thus, the domain of [tex]\((f \circ g)(x)\)[/tex] is all real numbers.
2. [tex]\( (g \circ f)(x) \)[/tex]:
The domain of [tex]\(f(x)\)[/tex] is all real numbers. The output of [tex]\(f(x)\)[/tex] must be within the domain of [tex]\(g(x)\)[/tex], which is all real numbers for [tex]\(x^2 + 4x - 3\)[/tex]. Thus, the domain of [tex]\((g \circ f)(x)\)[/tex] is all real numbers.
3. [tex]\( (f \circ f)(x) \)[/tex]:
The domain of [tex]\(f(x)\)[/tex] is all real numbers. The output of [tex]\(f(x)\)[/tex] must be within the domain of itself, which is all real numbers for [tex]\(4x^3\)[/tex]. Thus, the domain of [tex]\((f \circ f)(x)\)[/tex] is all real numbers.
So, the domains for [tex]\((f \circ g)(x)\)[/tex], [tex]\((g \circ f)(x)\)[/tex], and [tex]\((f \circ f)(x)\)[/tex] are all real numbers, which can be written as:
[tex]\[ \text{Domain: } (-\infty, \infty) \][/tex]
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