Domain of a Square Root Function

Oct 13, 2025 by miodeya.com 76 views

In mathematics, the domain of a function is the set of all possible input 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. In this article, we will explore the concept of domain and determine which function has a domain of ${x \mid x \geq 8}$ .

What is the Domain of a Function?

The domain of a function is an essential concept in mathematics, and it is used to determine the set of all possible input values for which the function is defined. A function is defined as a relation between a set of inputs, called the domain, and a set of possible outputs. The domain of a function is the set of all possible input values that can be plugged into the function without causing any problems.

Types of Domains

There are two types of domains: finite and infinite. A finite domain is a set of a finite number of elements, while an infinite domain is a set of an infinite number of elements. For example, the domain of the function $f(x) = x^2$ is the set of all real numbers, which is an infinite domain.

Domain of a Square Root Function

The square root function is a common function that is used in mathematics. The domain of a square root function is the set of all non-negative real numbers. In other words, the domain of the function $f(x) = \sqrt{x}$ is the set of all real numbers $x$ such that $x \geq 0$ .

Domain of the Given Functions

Now, let's analyze the given functions and determine which one has a domain of $\{x \mid x \geq 8\}$ .

A. $f(x)=\sqrt{x-8}+1$

The domain of this function is the set of all real numbers $x$ such that $x-8 \geq 0$ . This means that the domain of this function is the set of all real numbers $x$ such that $x \geq 8$ . Therefore, the domain of this function is $\{x \mid x \geq 8\}$ .

B. $f(x)=\sqrt{x+8}-1$

The domain of this function is the set of all real numbers $x$ such that $x+8 \geq 0$ . This means that the domain of this function is the set of all real numbers $x$ such that $x \geq -8$ . Therefore, the domain of this function is not $\{x \mid x \geq 8\}$ .

C. $f(x)=\sqrt{x-1}+8$

The domain of this function is the set of all real numbers $x$ such that $x-1 \geq 0$ . This means that the domain of this function is the set of all real numbers $x$ such that $x \geq 1$ . Therefore, the domain of this function is not $\{x \mid x \geq 8\}$ .

D. $f(x)=\sqrt{x+1}-8$

The domain of this function is the set of all real numbers $x$ such that $x+1 \geq 0$ . This means that the domain of this function is the set of all real numbers $x$ such that $x \geq -1$ . Therefore, the domain of this function is not $\{x \mid x \geq 8\}$ .

Conclusion

In conclusion, the function that has a domain of $\{x \mid x \geq 8\}$ is the function $f(x)=\sqrt{x-8}+1$ . This function is defined for all real numbers $x$ such that $x \geq 8$ , which is the set of all real numbers $x$ such that $x-8 \geq 0$ .

Final Answer

In the previous article, we discussed the concept of domain and determined which function has a domain of $\{x \mid x \geq 8\}$ . In this article, we will answer some frequently asked questions about the domain of a function.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input 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.

Q: What are the types of domains?

A: There are two types of domains: finite and infinite. A finite domain is a set of a finite number of elements, while an infinite domain is a set of an infinite number of elements.

Q: What is the domain of a square root function?

A: The domain of a square root function is the set of all non-negative real numbers. In other words, the domain of the function $f(x) = \sqrt{x}$ is the set of all real numbers $x$ such that $x \geq 0$ .

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

A: To determine the domain of a function, you need to identify any restrictions on the input values. For example, if the function involves a square root, you need to ensure that the input value is non-negative. If the function involves a fraction, you need to ensure that the denominator is not equal to zero.

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, while the range of a function is the set of all possible output values. In other words, the domain is the set of all possible x-values, while the range is the set of all possible y-values.

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

A: Yes, a function can have a domain of all real numbers. For example, the function $f(x) = x^2$ has a domain of all real numbers, since any real number can be squared.

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

A: Yes, a function can have a domain of a single value. For example, the function $f(x) = x$ has a domain of a single value, since the input value is fixed.

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 input values and graph the function accordingly. For example, if the function involves a square root, you need to ensure that the input value is non-negative.

Q: Can a function have a domain of a set of discrete values?

A: Yes, a function can have a domain of a set of discrete values. For example, the function $f(x) = x^2$ has a domain of a set of discrete values, since the input values are restricted to a set of specific values.

Conclusion

In conclusion, the domain of a function is an essential concept in mathematics, and it is used to determine the set of all possible input values for which the function is defined. By understanding the domain of a function, you can identify any restrictions on the input values and graph the function accordingly.

Final Answer

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