To determine the domain of [tex]\((b \circ a)(x)\)[/tex], we first need to understand the behavior of the functions [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex].
1. Define the Functions:
- [tex]\(a(x) = 3x + 1\)[/tex]
- [tex]\(b(x) = \sqrt{x - 4}\)[/tex]
2. Composite Function [tex]\((b \circ a)(x)\)[/tex]:
To find [tex]\((b \circ a)(x)\)[/tex], substitute [tex]\(a(x)\)[/tex] into [tex]\(b(x)\)[/tex]:
[tex]\[
(b \circ a)(x) = b(a(x)) = b(3x + 1) = \sqrt{3x + 1 - 4}
\][/tex]
Simplifying inside the square root:
[tex]\[
(b \circ a)(x) = \sqrt{3x - 3}
\][/tex]
3. Determine the Domain:
The expression inside the square root, [tex]\(3x - 3\)[/tex], must be non-negative for the square root to be defined (since the square root function only accepts non-negative arguments). Therefore, we need:
[tex]\[
3x - 3 \geq 0
\][/tex]
4. Solve the Inequality:
Add 3 to both sides of the inequality:
[tex]\[
3x \geq 3
\][/tex]
Divide both sides by 3:
[tex]\[
x \geq 1
\][/tex]
5. Conclusion:
The domain of the composite function [tex]\((b \circ a)(x)\)[/tex] is all [tex]\(x\)[/tex] such that [tex]\(x \geq 1\)[/tex].
So, the domain of [tex]\((b \circ a)(x)\)[/tex] is [tex]\([1, \infty)\)[/tex].
Answer: [tex]\([1, \infty)\)[/tex]
