To determine the domain of the function [tex]\( (b \circ a)(x) \)[/tex], where [tex]\( (b \circ a)(x) = b(a(x)) \)[/tex], we need to find when [tex]\( b \)[/tex] applied to [tex]\( a(x) \)[/tex] is defined. Let's break down the steps in detail.
1. Understand the functions involved:
- [tex]\( a(x) = 3x + 1 \)[/tex]
- [tex]\( b(x) = \sqrt{x - 4} \)[/tex]
2. Determine the composition [tex]\( (b \circ a)(x) = b(a(x)) \)[/tex]:
- Substituting [tex]\( a(x) \)[/tex] into [tex]\( b \)[/tex], we get:
[tex]\[
(b \circ a)(x) = b(3x + 1)
\][/tex]
3. Examine the function [tex]\( b(3x + 1) \)[/tex]:
- [tex]\( b(y) = \sqrt{y - 4} \)[/tex], so [tex]\( b(3x + 1) = \sqrt{(3x + 1) - 4} \)[/tex].
4. Simplify the expression under the square root:
[tex]\[
b(3x + 1) = \sqrt{3x + 1 - 4} = \sqrt{3x - 3}
\][/tex]
5. Set the condition for the square root to be defined:
- The expression inside the square root must be non-negative:
[tex]\[
3x - 3 \geq 0
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
- Add 3 to both sides:
[tex]\[
3x \geq 3
\][/tex]
- Divide both sides by 3:
[tex]\[
x \geq 1
\][/tex]
Hence, for the composition [tex]\( (b \circ a)(x) \)[/tex] to be defined, [tex]\( x \)[/tex] must be greater than or equal to 1.
Conclusion:
The domain of [tex]\( (b \circ a)(x) \)[/tex] is:
[tex]\[
x \geq 1
\][/tex]
