To determine the domain of the composite function [tex]\((b \circ a)(x)\)[/tex], which means [tex]\(b(a(x))\)[/tex], we'll follow these steps:
1. Understand the individual functions and their domains:
- The function [tex]\(a(x) = 3x + 1\)[/tex] is a linear function. A linear function is defined for all real numbers, so the domain of [tex]\(a(x)\)[/tex] is [tex]\((-\infty, \infty)\)[/tex].
- The function [tex]\(b(x) = \sqrt{x - 4}\)[/tex] is a square root function. For the square root function to be real-valued, the expression inside the square root must be non-negative. Thus, the domain of [tex]\(b(x)\)[/tex] is [tex]\(x \geq 4\)[/tex] or [tex]\([4, \infty)\)[/tex].
2. Determine the domain of [tex]\(a(x)\)[/tex] so that [tex]\(b(a(x))\)[/tex] is defined:
- In the composite function [tex]\(b(a(x))\)[/tex], the output of [tex]\(a(x)\)[/tex] (which is [tex]\(3x + 1\)[/tex]) becomes the input of [tex]\(b(x)\)[/tex]. So, we need [tex]\(3x + 1\)[/tex] to be within the domain of [tex]\(b(x)\)[/tex].
- Since [tex]\(b(x)\)[/tex] is defined for [tex]\(x \geq 4\)[/tex], we need:
[tex]\[
3x + 1 \geq 4
\][/tex]
3. Solve the inequality [tex]\(3x + 1 \geq 4\)[/tex]:
- Subtract 1 from both sides:
[tex]\[
3x \geq 3
\][/tex]
- Divide both sides by 3:
[tex]\[
x \geq 1
\][/tex]
4. Conclusion:
- The domain of the composite function [tex]\( (b \circ a)(x) \)[/tex] includes all [tex]\(x\)[/tex] such that [tex]\(x \geq 1\)[/tex].
- Therefore, the domain of [tex]\( (b \circ a)(x) \)[/tex] is [tex]\( [1, \infty) \)[/tex].
Thus, the correct answer is [tex]\( [1, \infty) \)[/tex].
