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Sagot :
To solve this problem, we'll find each composite function step-by-step and then determine the domains of each.
### (a) [tex]\( f \circ g \)[/tex]
First, recall the given functions:
[tex]\[ f(x) = x^2 \][/tex]
[tex]\[ g(x) = x^2 + 9 \][/tex]
The composite function [tex]\( f \circ g \)[/tex] means we need to substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f(x^2 + 9) \][/tex]
Since [tex]\( f(x) = x^2 \)[/tex], we replace the [tex]\( x \)[/tex] in [tex]\( f \)[/tex] with [tex]\( x^2 + 9 \)[/tex]:
[tex]\[ f(g(x)) = (x^2 + 9)^2 \][/tex]
So,
[tex]\[ f \circ g = (x^2 + 9)^2 \][/tex]
Domain:
Both functions [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = x^2 + 9 \)[/tex] are defined for all real numbers. Therefore, the domain of [tex]\( f \circ g \)[/tex] is all real numbers:
[tex]\[ \text{Domain of } f \circ g = \mathbb{R} \][/tex]
### (b) [tex]\( g \circ f \)[/tex]
The composite function [tex]\( g \circ f \)[/tex] means we need to substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g(x^2) \][/tex]
Since [tex]\( g(x) = x^2 + 9 \)[/tex], we replace the [tex]\( x \)[/tex] in [tex]\( g \)[/tex] with [tex]\( x^2 \)[/tex]:
[tex]\[ g(f(x)) = (x^2)^2 + 9 \][/tex]
[tex]\[ g(f(x)) = x^4 + 9 \][/tex]
So,
[tex]\[ g \circ f = x^4 + 9 \][/tex]
Domain:
Similarly, both functions [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = x^2 + 9 \)[/tex] are defined for all real numbers. Therefore, the domain of [tex]\( g \circ f \)[/tex] is all real numbers:
[tex]\[ \text{Domain of } g \circ f = \mathbb{R} \][/tex]
### (c) [tex]\( f \circ f \)[/tex]
The composite function [tex]\( f \circ f \)[/tex] means we need to substitute [tex]\( f(x) \)[/tex] into itself:
[tex]\[ f(f(x)) = f(x^2) \][/tex]
Since [tex]\( f(x) = x^2 \)[/tex], we replace the [tex]\( x \)[/tex] in [tex]\( f \)[/tex] with [tex]\( x^2 \)[/tex]:
[tex]\[ f(f(x)) = (x^2)^2 \][/tex]
[tex]\[ f(f(x)) = x^4 \][/tex]
So,
[tex]\[ f \circ f = x^4 \][/tex]
Domain:
Since [tex]\( f(x) = x^2 \)[/tex] is defined for all real numbers, substituting it back into itself does not change this. Therefore, the domain of [tex]\( f \circ f \)[/tex] is all real numbers:
[tex]\[ \text{Domain of } f \circ f = \mathbb{R} \][/tex]
### (d) [tex]\( g \circ g \)[/tex]
The composite function [tex]\( g \circ g \)[/tex] means we need to substitute [tex]\( g(x) \)[/tex] into itself:
[tex]\[ g(g(x)) = g(x^2 + 9) \][/tex]
Since [tex]\( g(x) = x^2 + 9 \)[/tex], we replace the [tex]\( x \)[/tex] in [tex]\( g \)[/tex] with [tex]\( x^2 + 9 \)[/tex]:
[tex]\[ g(g(x)) = (x^2 + 9)^2 + 9 \][/tex]
So,
[tex]\[ g \circ g = (x^2 + 9)^2 + 9 \][/tex]
Domain:
Since [tex]\( g(x) = x^2 + 9 \)[/tex] is defined for all real numbers, substituting it back into itself does not change this. Therefore, the domain of [tex]\( g \circ g \)[/tex] is all real numbers:
[tex]\[ \text{Domain of } g \circ g = \mathbb{R} \][/tex]
### Summary of Results:
(a) [tex]\( f \circ g = (x^2 + 9)^2 \)[/tex]
[tex]\[ \text{Domain of } f \circ g = \mathbb{R} \][/tex]
(b) [tex]\( g \circ f = x^4 + 9 \)[/tex]
[tex]\[ \text{Domain of } g \circ f = \mathbb{R} \][/tex]
(c) [tex]\( f \circ f = x^4 \)[/tex]
[tex]\[ \text{Domain of } f \circ f = \mathbb{R} \][/tex]
(d) [tex]\( g \circ g = (x^2 + 9)^2 + 9 \)[/tex]
[tex]\[ \text{Domain of } g \circ g = \mathbb{R} \][/tex]
### (a) [tex]\( f \circ g \)[/tex]
First, recall the given functions:
[tex]\[ f(x) = x^2 \][/tex]
[tex]\[ g(x) = x^2 + 9 \][/tex]
The composite function [tex]\( f \circ g \)[/tex] means we need to substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f(x^2 + 9) \][/tex]
Since [tex]\( f(x) = x^2 \)[/tex], we replace the [tex]\( x \)[/tex] in [tex]\( f \)[/tex] with [tex]\( x^2 + 9 \)[/tex]:
[tex]\[ f(g(x)) = (x^2 + 9)^2 \][/tex]
So,
[tex]\[ f \circ g = (x^2 + 9)^2 \][/tex]
Domain:
Both functions [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = x^2 + 9 \)[/tex] are defined for all real numbers. Therefore, the domain of [tex]\( f \circ g \)[/tex] is all real numbers:
[tex]\[ \text{Domain of } f \circ g = \mathbb{R} \][/tex]
### (b) [tex]\( g \circ f \)[/tex]
The composite function [tex]\( g \circ f \)[/tex] means we need to substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g(x^2) \][/tex]
Since [tex]\( g(x) = x^2 + 9 \)[/tex], we replace the [tex]\( x \)[/tex] in [tex]\( g \)[/tex] with [tex]\( x^2 \)[/tex]:
[tex]\[ g(f(x)) = (x^2)^2 + 9 \][/tex]
[tex]\[ g(f(x)) = x^4 + 9 \][/tex]
So,
[tex]\[ g \circ f = x^4 + 9 \][/tex]
Domain:
Similarly, both functions [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = x^2 + 9 \)[/tex] are defined for all real numbers. Therefore, the domain of [tex]\( g \circ f \)[/tex] is all real numbers:
[tex]\[ \text{Domain of } g \circ f = \mathbb{R} \][/tex]
### (c) [tex]\( f \circ f \)[/tex]
The composite function [tex]\( f \circ f \)[/tex] means we need to substitute [tex]\( f(x) \)[/tex] into itself:
[tex]\[ f(f(x)) = f(x^2) \][/tex]
Since [tex]\( f(x) = x^2 \)[/tex], we replace the [tex]\( x \)[/tex] in [tex]\( f \)[/tex] with [tex]\( x^2 \)[/tex]:
[tex]\[ f(f(x)) = (x^2)^2 \][/tex]
[tex]\[ f(f(x)) = x^4 \][/tex]
So,
[tex]\[ f \circ f = x^4 \][/tex]
Domain:
Since [tex]\( f(x) = x^2 \)[/tex] is defined for all real numbers, substituting it back into itself does not change this. Therefore, the domain of [tex]\( f \circ f \)[/tex] is all real numbers:
[tex]\[ \text{Domain of } f \circ f = \mathbb{R} \][/tex]
### (d) [tex]\( g \circ g \)[/tex]
The composite function [tex]\( g \circ g \)[/tex] means we need to substitute [tex]\( g(x) \)[/tex] into itself:
[tex]\[ g(g(x)) = g(x^2 + 9) \][/tex]
Since [tex]\( g(x) = x^2 + 9 \)[/tex], we replace the [tex]\( x \)[/tex] in [tex]\( g \)[/tex] with [tex]\( x^2 + 9 \)[/tex]:
[tex]\[ g(g(x)) = (x^2 + 9)^2 + 9 \][/tex]
So,
[tex]\[ g \circ g = (x^2 + 9)^2 + 9 \][/tex]
Domain:
Since [tex]\( g(x) = x^2 + 9 \)[/tex] is defined for all real numbers, substituting it back into itself does not change this. Therefore, the domain of [tex]\( g \circ g \)[/tex] is all real numbers:
[tex]\[ \text{Domain of } g \circ g = \mathbb{R} \][/tex]
### Summary of Results:
(a) [tex]\( f \circ g = (x^2 + 9)^2 \)[/tex]
[tex]\[ \text{Domain of } f \circ g = \mathbb{R} \][/tex]
(b) [tex]\( g \circ f = x^4 + 9 \)[/tex]
[tex]\[ \text{Domain of } g \circ f = \mathbb{R} \][/tex]
(c) [tex]\( f \circ f = x^4 \)[/tex]
[tex]\[ \text{Domain of } f \circ f = \mathbb{R} \][/tex]
(d) [tex]\( g \circ g = (x^2 + 9)^2 + 9 \)[/tex]
[tex]\[ \text{Domain of } g \circ g = \mathbb{R} \][/tex]
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