If $a(x) = 3x + 1$ And $b(x) = \sqrt{x - 4}$, What Is The Domain Of \$(b \circ A)(x)$[/tex\]?A. $(-\infty, \infty$\]B. \[0, \infty$\]C. \[1, \infty$\]D. \[4, \infty$\]
In mathematics, the composition of functions is a fundamental concept that allows us to combine two or more functions to create a new function. Given two functions, $a(x)$ and $b(x)$, the composition of $b$ and $a$, denoted as $(b \circ a)(x)$, is defined as $b(a(x))$. In this article, we will explore the composition of two given functions, $a(x) = 3x + 1$ and $b(x) = \sqrt{x - 4}$, and determine the domain of the resulting function $(b \circ a)(x)$.
The Composition of Functions
To find the composition of $b$ and $a$, we need to substitute $a(x)$ into $b(x)$ in place of $x$. This gives us:
Simplifying the expression, we get:
The Domain of the Composition
To determine the domain of the composition $(b \circ a)(x)$, we need to consider the restrictions imposed by both functions $a(x)$ and $b(x)$. The function $a(x) = 3x + 1$ is defined for all real numbers $x$, as it is a linear function. However, the function $b(x) = \sqrt{x - 4}$ is only defined for $x \geq 4$, as the square root of a negative number is not a real number.
Since the composition $(b \circ a)(x)$ involves the function $b(x)$, which is only defined for $x \geq 4$, we need to ensure that the input to $b(x)$, which is $a(x)$, is also greater than or equal to 4. This means that we need to find the values of $x$ for which $a(x) \geq 4$.
Finding the Values of x
To find the values of $x$ for which $a(x) \geq 4$, we can set up the inequality:
Subtracting 1 from both sides, we get:
Dividing both sides by 3, we get:
Therefore, the values of $x$ for which $a(x) \geq 4$ are $x \geq 1$.
The Domain of the Composition
Since the composition $(b \circ a)(x)$ involves the function $b(x)$, which is only defined for $x \geq 4$, and the values of $x$ for which $a(x) \geq 4$ are $x \geq 1$, we can conclude that the domain of the composition $(b \circ a)(x)$ is:
However, we need to consider the values of $x$ for which $a(x)$ is defined, which is all real numbers $x$. Therefore, the domain of the composition $(b \circ a)(x)$ is:
Conclusion
In conclusion, the domain of the composition $(b \circ a)(x)$ is $x \geq 4$. This means that the function $(b \circ a)(x)$ is only defined for values of $x$ greater than or equal to 4.
Answer
The correct answer is:
D. $[4, \infty)$
Final Thoughts
In our previous article, we explored the composition of two given functions, $a(x) = 3x + 1$ and $b(x) = \sqrt{x - 4}$, and determined the domain of the resulting function $(b \circ a)(x)$. In this article, we will answer some frequently asked questions related to the composition of functions and its domain.
Q: What is the composition of functions?
A: The composition of functions is a way of combining two or more functions to create a new function. Given two functions, $a(x)$ and $b(x)$, the composition of $b$ and $a$, denoted as $(b \circ a)(x)$, is defined as $b(a(x))$.
Q: How do I find the composition of two functions?
A: To find the composition of two functions, you need to substitute the first function into the second function in place of $x$. For example, if we want to find the composition of $a(x) = 3x + 1$ and $b(x) = \sqrt{x - 4}$, we would substitute $a(x)$ into $b(x)$ in place of $x$.
Q: What is the domain of the composition of two functions?
A: The domain of the composition of two functions is the set of all possible input values for which the composition is defined. In other words, it is the set of all possible values of $x$ for which the composition $(b \circ a)(x)$ is defined.
Q: How do I determine the domain of the composition of two functions?
A: To determine the domain of the composition of two functions, you need to consider the restrictions imposed by both functions. You need to find the values of $x$ for which the input to the second function is defined, and then find the values of $x$ for which the output of the first function is within the domain of the second function.
Q: What is the difference between the domain and the range of a function?
A: The domain of a function is the set of all possible input values for which the function is defined. The range of a function is the set of all possible output values of the function.
Q: Can the domain of a function be empty?
A: Yes, the domain of a function can be empty. This means that there are no possible input values for which the function is defined.
Q: Can the range of a function be empty?
A: No, the range of a function cannot be empty. This means that there is at least one possible output value of the function.
Q: How do I graph the composition of two functions?
A: To graph the composition of two functions, you need to first graph the first function, and then graph the second function with the output of the first function as the input.
Q: Can the composition of two functions be a one-to-one function?
A: Yes, the composition of two functions can be a one-to-one function. This means that the function has a unique output value for each input value.
Q: Can the composition of two functions be a many-to-one function?
A: Yes, the composition of two functions can be a many-to-one function. This means that the function has multiple output values for each input value.
Q: Can the composition of two functions be a one-to-many function?
A: No, the composition of two functions cannot be a one-to-many function. This means that the function has a unique output value for each input value.
Q: Can the composition of two functions be a many-to-many function?
A: Yes, the composition of two functions can be a many-to-many function. This means that the function has multiple output values for each input value.
Conclusion
In conclusion, the composition of functions is a powerful tool in mathematics that allows us to combine two or more functions to create a new function. The domain of the composition of two functions is the set of all possible input values for which the composition is defined. We hope that this Q&A article has helped to clarify any questions you may have had about the composition of functions and its domain.