What Is The Domain Of The Following Function: F ( X ) = X + 2 X − 7 F(x)=\frac{\sqrt{x+2}}{x-7} F ( X ) = X − 7 X + 2 A. X ≠ 7 X \neq 7 X  = 7 B. {-2,7) \cup (7, \infty }$ C. ( 7 , ∞ (7, \infty ( 7 , ∞ ] D. {-2, \infty }$ E. All Real Numbers

Mar 13, 2025 by ADMIN 250 views

$What is the Domain of the Function $f(x)=\frac{\sqrt{x+2}}{x-7}$?$

Understanding the Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it is the set of all possible values of x that can be plugged into the function without causing any problems, such as division by zero or taking the square root of a negative number.

Identifying the Restrictions on the Domain

To find the domain of the function $f(x)=\frac{\sqrt{x+2}}{x-7}$ , we need to identify any restrictions on the domain. There are two main restrictions to consider:

Square Root Restriction: The expression inside the square root must be non-negative, i.e., $x+2 \geq 0$ . This means that $x \geq -2$ .
Division by Zero Restriction: The denominator of the fraction cannot be zero, i.e., $x-7 \neq 0$ . This means that $x \neq 7$ .

Combining the Restrictions

Now that we have identified the two restrictions, we need to combine them to find the domain of the function. The domain is the set of all values of x that satisfy both restrictions.

The square root restriction requires that $x \geq -2$ .
The division by zero restriction requires that $x \neq 7$ .

Since the two restrictions are not mutually exclusive, we need to find the intersection of the two sets of values. The intersection of $x \geq -2$ and $x \neq 7$ is $[-2, 7) \cup (7, \infty)$ .

Conclusion

The domain of the function $f(x)=\frac{\sqrt{x+2}}{x-7}$ is $[-2, 7) \cup (7, \infty)$ . This means that the function is defined for all values of x except $x = 7$ .

Comparison with the Options

Now that we have found the domain of the function, we can compare it with the options provided:

Option A: $x \neq 7$ is a part of the domain, but it is not the entire domain.
Option B: $[-2, 7) \cup (7, \infty)$ is the correct domain.
Option C: $(7, \infty)$ is not the correct domain, as the function is also defined for $x \in [-2, 7)$ .
Option D: $[-2, \infty)$ is not the correct domain, as the function is not defined for $x = 7$ .
Option E: All real numbers is not the correct domain, as the function is not defined for $x = 7$ .

The correct answer is B. $[-2, 7) \cup (7, \infty)$ .

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined.

Q: Why is it important to find the domain of a function?

A: Finding the domain of a function is important because it helps us to determine the values of x for which the function is defined. This is crucial in understanding the behavior of the function and making predictions about its output.

Q: How do I find the domain of a function?

A: To find the domain of a function, you need to identify any restrictions on the domain. These restrictions can include:

Square root restrictions: The expression inside the square root must be non-negative.
Division by zero restrictions: The denominator of the fraction cannot be zero.
Other restrictions: Such as the function being undefined at certain points.

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 (x-values) for which the function is defined. The range of a function is the set of all possible output values (y-values) that the function can produce.

Q: Can a function have a domain of all real numbers?

A: Yes, a function can have a domain of all real numbers. This means that the function is defined for all possible values of x.

Q: Can a function have a domain of a single point?

A: Yes, a function can have a domain of a single point. This means that the function is defined only at that specific point.

Q: How do I determine if a function is defined at a particular point?

A: To determine if a function is defined at a particular point, you need to check if the function satisfies the restrictions on the domain. If the function satisfies the restrictions, then it is defined at that point.

Q: What is the significance of the domain of a function in real-world applications?

A: The domain of a function is significant in real-world applications because it helps us to understand the limitations and constraints of the function. For example, in physics, the domain of a function can represent the range of possible values of a variable, such as temperature or pressure.

Q: Can the domain of a function change depending on the context?

A: Yes, the domain of a function can change depending on the context. For example, in mathematics, the domain of a function may be all real numbers, but in a specific application, the domain may be restricted to a subset of real numbers.

Q: How do I graph a function with a restricted domain?

A: To graph a function with a restricted domain, you need to identify the restrictions on the domain and then graph the function accordingly. For example, if the function is defined only for x > 0, then you would graph the function only for x > 0.

Q: Can a function have multiple domains?

A: No, a function cannot have multiple domains. The domain of a function is a set of values, and a set cannot have multiple values.

Q: How do I determine if a function is continuous or discontinuous?

A: To determine if a function is continuous or discontinuous, you need to check if the function satisfies the conditions for continuity. A function is continuous if it is defined at all points in its domain and if the limit of the function as x approaches a point is equal to the value of the function at that point.

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

A: The domain of a function is related to the graph of the function in that the graph of the function is only defined for the values of x in the domain of the function.

Q: Can the domain of a function be changed by a transformation?

A: Yes, the domain of a function can be changed by a transformation. For example, if a function is stretched or compressed, its domain may also be stretched or compressed.

Q: How do I determine if a function is one-to-one or many-to-one?

A: To determine if a function is one-to-one or many-to-one, you need to check if the function satisfies the conditions for one-to-one or many-to-one functions. A function is one-to-one if it passes the horizontal line test, and a function is many-to-one if it does not pass the horizontal line test.

Q: What is the significance of the domain of a function in calculus?

A: The domain of a function is significant in calculus because it helps us to understand the behavior of the function and its derivatives. For example, in finding the derivative of a function, we need to know the domain of the function to ensure that the derivative is defined.

Q: Can the domain of a function be changed by a composition of functions?

A: Yes, the domain of a function can be changed by a composition of functions. For example, if two functions are composed, the domain of the resulting function may be different from the domains of the individual functions.

Q: How do I determine if a function is invertible?

A: To determine if a function is invertible, you need to check if the function satisfies the conditions for invertibility. A function is invertible if it is one-to-one and if its inverse is also a function.

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

A: The domain of a function is related to the inverse of the function in that the domain of the inverse function is the range of the original function, and the range of the inverse function is the domain of the original function.