To find the domain of the composition [tex]\((b \circ a)(x)\)[/tex], where [tex]\(a(x) = 3x + 1\)[/tex] and [tex]\(b(x) = \sqrt{x - 4}\)[/tex], we need to ensure that the function [tex]\(b(a(x))\)[/tex] is defined. This involves two steps:
1. Determine the range of [tex]\(a(x)\)[/tex].
2. Determine the values of [tex]\(x\)[/tex] for which [tex]\(b(a(x))\)[/tex] is defined.
Let's start with the first step:
[tex]\[ a(x) = 3x + 1 \][/tex]
Next, we need to consider the domain of [tex]\(b(x) = \sqrt{x - 4}\)[/tex]. For [tex]\(b(x)\)[/tex] to be defined, the expression inside the square root must be non-negative:
[tex]\[ x - 4 \geq 0 \][/tex]
This simplifies to:
[tex]\[ x \geq 4 \][/tex]
Now, we need to ensure that the input to [tex]\(b\)[/tex] (which in this case is [tex]\(a(x)\)[/tex]) respects this condition:
[tex]\[ a(x) \geq 4 \][/tex]
Substitute [tex]\(a(x)\)[/tex] with [tex]\(3x + 1\)[/tex]:
[tex]\[ 3x + 1 \geq 4 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ 3x + 1 - 4 \geq 0 \][/tex]
[tex]\[ 3x - 3 \geq 0 \][/tex]
[tex]\[ x \geq 1 \][/tex]
Therefore, for [tex]\(b(a(x))\)[/tex] to be defined, [tex]\(x\)[/tex] must satisfy:
[tex]\[ x \geq 1 \][/tex]
Thus, the domain of [tex]\((b \circ a)(x)\)[/tex] is:
[tex]\[ [1, \infty) \][/tex]
So, the correct answer is:
[tex]\[ [1, \infty) \][/tex]
