Select The Correct Answer.If F ( X ) = 2 − X 1 2 F(x) = 2 - X^{\frac{1}{2}} F ( X ) = 2 − X 2 1 ​ And G ( X ) = X 2 − 9 G(x) = X^2 - 9 G ( X ) = X 2 − 9 , What Is The Domain Of G ( X ) + F ( X G(x) + F(x G ( X ) + F ( X ]?A. ( − ∞ , 2 ] ∪ [ 2 , ∞ (-\infty, 2] \cup [2, \infty ( − ∞ , 2 ] ∪ [ 2 , ∞ ]B. {0, 4) \cup (4, \infty }$C. $(-\infty, 2] \cup [4,

When dealing with composite functions, it's essential to understand the domain of each individual function involved. In this case, we have two functions: $f(x) = 2 - x^{\frac{1}{2}}$ and $g(x) = x^2 - 9$. We're asked to find the domain of the composite function $g(x) + f(x)$.

The Domain of $f(x)$

The function $f(x) = 2 - x^{\frac{1}{2}}$ involves a square root, which means we need to consider the values of $x$ that make the expression under the square root non-negative. In other words, we need to find the values of $x$ such that $x^{\frac{1}{2}} \geq 0$.

Since the square root of any real number is non-negative, we can conclude that $x^{\frac{1}{2}} \geq 0$ for all real numbers $x$. However, we also need to consider the values of $x$ that make the expression under the square root undefined. In this case, the expression is undefined when $x < 0$, since the square root of a negative number is undefined in the real number system.

Therefore, the domain of $f(x)$ is all real numbers $x$ such that $x \geq 0$.

The Domain of $g(x)$

The function $g(x) = x^2 - 9$ involves a quadratic expression, which is defined for all real numbers $x$. However, we need to consider the values of $x$ that make the expression under the square root in $f(x)$ undefined. In this case, the expression is undefined when $x < 0$, since the square root of a negative number is undefined in the real number system.

Therefore, the domain of $g(x)$ is all real numbers $x$.

The Domain of $g(x) + f(x)$

To find the domain of the composite function $g(x) + f(x)$, we need to consider the values of $x$ that make both $g(x)$ and $f(x)$ defined. Since the domain of $g(x)$ is all real numbers $x$, and the domain of $f(x)$ is all real numbers $x$ such that $x \geq 0$, the domain of $g(x) + f(x)$ is the intersection of these two domains.

Therefore, the domain of $g(x) + f(x)$ is all real numbers $x$ such that $x \geq 0$.

Solving the Problem

Now that we have found the domain of $g(x) + f(x)$, we can solve the problem by finding the values of $x$ that satisfy this domain.

Let's consider the two possible cases:

• Case 1: $x \geq 0$
• Case 2: $x < 0$

For Case 1, we have $x \geq 0$, which means that $x$ is non-negative. In this case, the expression $g(x) + f(x)$ is defined for all non-negative values of $x$.

For Case 2, we have $x < 0$, which means that $x$ is negative. In this case, the expression $g(x) + f(x)$ is undefined, since the square root of a negative number is undefined in the real number system.

Therefore, the domain of $g(x) + f(x)$ is all real numbers $x$ such that $x \geq 0$.

Conclusion

In conclusion, the domain of the composite function $g(x) + f(x)$ is all real numbers $x$ such that $x \geq 0$. This means that the correct answer is:

A. $(-\infty, 2] \cup [2, \infty)$

This answer is correct because it represents the domain of the composite function $g(x) + f(x)$, which is all real numbers $x$ such that $x \geq 0$.

Final Answer

In our previous article, we discussed the domain of composite functions and how to find the domain of the composite function $g(x) + f(x)$. In this article, we'll answer some frequently asked questions about the 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 function is defined. In other words, it's the set of all possible values of $x$ for which the 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 consider the domains of the individual functions involved. You need to find the intersection of the domains of the individual functions to get the domain of the composite function.

Q: What is the intersection of two sets?

A: The intersection of two sets is the set of all elements that are common to both sets. In other words, it's the set of all elements that are present in both sets.

Q: How do I find the intersection of two sets?

A: To find the intersection of two sets, you need to identify the elements that are common to both sets. You can do this by listing out the elements of both sets and then identifying the elements that are present in both lists.

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 for which the function is defined.

Q: How do I find the range of a composite function?

A: To find the range of a composite function, you need to consider the range of the individual functions involved. You need to find the intersection of the ranges of the individual functions to get the range of the composite function.

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 function is defined. If the domain of a composite function is not properly defined, it can lead to incorrect results or even undefined behavior.

Q: Can I have a negative value in the domain of a composite function?

A: Yes, you can have a negative value in the domain of a composite function. However, you need to ensure that the negative value is within the domain of the individual functions involved.

Q: How do I determine if a value is within the domain of a composite function?

A: To determine if a value is within the domain of a composite function, you need to check if the value is within the domain of the individual functions involved. You can do this by checking if the value satisfies the conditions of the individual functions.

Q: What is the relationship between the domain and the range of a composite function?

A: The domain and the range of a composite function are related in the sense that the domain of the composite function determines the set of all possible input values for which the function is defined, and the range of the composite function determines the set of all possible output values for which the function is defined.

Conclusion

In conclusion, the domain of a composite function is the set of all possible input values for which the function is defined. To find the domain of a composite function, you need to consider the domains of the individual functions involved and find the intersection of the domains. The domain of a composite function is significant because it determines the set of all possible input values for which the function is defined.

Final Answer

The final answer is that the domain of a composite function is the set of all possible input values for which the function is defined.