If $a(x) = 3x + 1$ And $b(x) = \sqrt{x - 4}$, What Is The Domain Of $(b \circ A)(x$\]?A. $(-\infty, \infty$\] B. \[0, \infty$\] C. \[1, \infty$\] D. \[4, \infty$\]
Introduction
In mathematics, composite functions are a crucial concept in understanding the behavior of functions. When two functions are composed together, the output of the first function becomes the input for the second function. In this article, we will explore the concept of composite functions and determine the domain of a composite function given two functions.
What are Composite Functions?
A composite function is a function that is formed by combining two or more functions. The output of the first function becomes the input for the second function. For example, if we have two functions f(x) and g(x), the composite function (g ∘ f)(x) is defined as g(f(x)).
Domain of Composite Functions
The domain of a composite function is the set of all possible input values for which the composite function is defined. In other words, it is the set of all possible values of x for which the composite function is valid.
Example: Domain of (b ∘ a)(x)
Let's consider the two functions:
a(x) = 3x + 1 b(x) = √(x - 4)
We want to find the domain of the composite function (b ∘ a)(x). To do this, we need to find the set of all possible values of x for which the composite function is defined.
Step 1: Find the Output of a(x)
The output of a(x) is 3x + 1. This is the input for the function b(x).
Step 2: Find the Input of b(x)
The input of b(x) is x - 4. Since the output of a(x) is 3x + 1, we can substitute this value into the input of b(x):
b(a(x)) = √((3x + 1) - 4)
Step 3: Simplify the Expression
Simplifying the expression, we get:
b(a(x)) = √(3x - 3)
Step 4: Determine the Domain
The domain of b(a(x)) is the set of all possible values of x for which the expression √(3x - 3) is defined. Since the square root of a negative number is not defined, we need to find the values of x for which 3x - 3 ≥ 0.
Solving the Inequality
Solving the inequality 3x - 3 ≥ 0, we get:
3x ≥ 3 x ≥ 1
Conclusion
Therefore, the domain of (b ∘ a)(x) is [1, ∞).
Answer
The correct answer is C. [1, ∞).
Conclusion
In conclusion, the domain of a composite function is the set of all possible input values for which the composite function is defined. By following the steps outlined in this article, we can determine the domain of a composite function given two functions. The domain of (b ∘ a)(x) is [1, ∞).
References
- [1] "Composite Functions" by Khan Academy
- [2] "Domain of a Composite Function" by Math Open Reference
Additional Resources
- [1] "Composite Functions" by Wolfram MathWorld
- [2] "Domain of a Composite Function" by Purplemath
Final Thoughts
Q&A: Domain of Composite Functions
Q: What is the domain of a composite function?
A: The domain of a composite function is the set of all possible input values for which the composite function is defined.
Q: How do I find the domain of a composite function?
A: To find the domain of a composite function, you need to follow these steps:
- Find the output of the first function.
- Find the input of the second function.
- Substitute the output of the first function into the input of the second function.
- Simplify the expression.
- Determine the domain by finding the values of x for which the expression is defined.
Q: What is the difference between the domain and range of a composite function?
A: The domain of a composite function is the set of all possible input values for which the composite function is defined. The range of a composite function is the set of all possible output values of the composite function.
Q: Can the domain of a composite function be empty?
A: Yes, the domain of a composite function can be empty. This occurs when the output of the first function is not in the domain of the second function.
Q: How do I determine if the domain of a composite function is empty?
A: To determine if the domain of a composite function is empty, you need to check if the output of the first function is in the domain of the second function. If it is not, then the domain of the composite function is empty.
Q: Can the domain of a composite function be infinite?
A: Yes, the domain of a composite function can be infinite. This occurs when the output of the first function is in the domain of the second function for all possible input values.
Q: How do I determine if the domain of a composite function is infinite?
A: To determine if the domain of a composite function is infinite, you need to check if the output of the first function is in the domain of the second function for all possible input values. If it is, then the domain of the composite function is infinite.
Q: What is the significance of the domain of a composite function?
A: The domain of a composite function is significant because it determines the set of all possible input values for which the composite function is defined. This is important because it helps us to understand the behavior of the composite function and to determine its range.
Q: Can the domain of a composite function be a subset of the domain of one of the functions?
A: Yes, the domain of a composite function can be a subset of the domain of one of the functions. This occurs when the output of the first function is in the domain of the second function, but not for all possible input values.
Q: How do I determine if the domain of a composite function is a subset of the domain of one of the functions?
A: To determine if the domain of a composite function is a subset of the domain of one of the functions, you need to check if the output of the first function is in the domain of the second function. If it is, then the domain of the composite function is a subset of the domain of one of the functions.
Conclusion
In conclusion, the domain of a composite function is the set of all possible input values for which the composite function is defined. By following the steps outlined in this article, we can determine the domain of a composite function and gain a deeper understanding of the behavior of functions.
References
- [1] "Composite Functions" by Khan Academy
- [2] "Domain of a Composite Function" by Math Open Reference
Additional Resources
- [1] "Composite Functions" by Wolfram MathWorld
- [2] "Domain of a Composite Function" by Purplemath
Final Thoughts
In this article, we explored the concept of composite functions and determined the domain of a composite function given two functions. By following the steps outlined in this article, we can determine the domain of a composite function and gain a deeper understanding of the behavior of functions.