### Given Functions:
- [tex]\( f(x) = x + 3 \)[/tex]
- [tex]\( g(x) = \sqrt{1 - x^2} \)[/tex]
### Composition of Functions:
1. Composition of functions involves applying one function to the result of another.
### (a) [tex]\((f \circ g)(x)\)[/tex]:
- This means [tex]\( f(g(x)) \)[/tex], which means we first apply [tex]\( g(x) \)[/tex] and then apply [tex]\( f \)[/tex] to the result of [tex]\( g(x) \)[/tex].
[tex]\[
(f \circ g)(x) = f(g(x))
\][/tex]
- Calculate [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = \sqrt{1 - x^2}
\][/tex]
- Now, substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(g(x)) = f(\sqrt{1 - x^2}) = \sqrt{1 - x^2} + 3
\][/tex]
Domain of [tex]\( (f \circ g)(x) \)[/tex]:
- For [tex]\( g(x) = \sqrt{1 - x^2} \)[/tex] to be defined, the argument of the square root must be non-negative:
[tex]\[
1 - x^2 \geq 0 \implies -1 \leq x \leq 1
\][/tex]
- Therefore, the domain of [tex]\( (f \circ g)(x) \)[/tex] is [tex]\( \left[-1, 1\right] \)[/tex].
### (b) [tex]\((g \circ 0)(x)\)[/tex]:
- This notation is unconventional. If it means [tex]\( g(0) \)[/tex]:
[tex]\[
g(0) = \sqrt{1 - 0^2} = \sqrt{1} = 1
\][/tex]
### (c) [tex]\((I \circ f)(x)\)[/tex]:
- The identity function [tex]\( I(x) = x \)[/tex].
- Therefore, [tex]\( (I \circ f)(x) \)[/tex] means [tex]\( I(f(x)) \)[/tex]:
[tex]\[
(I \circ f)(x) = I(f(x)) = I(x + 3) = x + 3
\][/tex]
Domain of [tex]\( (I \circ f)(x) \)[/tex]:
- The domain of [tex]\( f(x) \)[/tex] is all real numbers [tex]\( \mathbb{R} \)[/tex].
- Since [tex]\( I \)[/tex] is also defined for all real numbers, the domain of [tex]\( (I \circ f)(x) \)[/tex] is [tex]\( \mathbb{R} \)[/tex].
### Correct Approach to Find Composition of Functions:
- Option A: [tex]\( (f \circ g)(x) = f(g(x)) \)[/tex].
- This is the correct definition of the composition of functions. You first apply [tex]\( g(x) \)[/tex] and then apply [tex]\( f \)[/tex] to the result of [tex]\( g(x) \)[/tex].
### Final Answers:
(a) [tex]\( (f \circ g)(x) = \sqrt{1 - x^2} + 3 \)[/tex]
- Domain: [tex]\([-1, 1]\)[/tex]
(b) [tex]\( g(0) = 1 \)[/tex]
(c) [tex]\( (I \circ f)(x) = x + 3 \)[/tex]
- Domain: [tex]\( \mathbb{R} \)[/tex]
Correct approach to find the composition of functions:
- Answer: A. [tex]\( (f \circ g)(x) = f(g(x)) \)[/tex]
