For [tex]\( f(x)=x^2 \)[/tex] and [tex]\( g(x)=x^2+9 \)[/tex], find the following composite functions and state the domain of each.

(a) [tex]\( f \circ g \)[/tex]

(b) [tex]\( g \circ f \)[/tex]

(c) [tex]\( f \circ f \)[/tex]

(d) [tex]\( g \circ g \)[/tex]



Answer :

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]