Restrictions Imposed by the Square Root Symbol

Oct 13, 2025 by magicalceylontours.com 125 views

Introduction to Domain of a Function

When dealing with functions, it's essential to understand the concept of the domain. 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's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output.

Understanding the Function p(x) = √(x-1) + 2

The given function is p(x) = √(x-1) + 2. To determine the domain of this function, we need to consider the restrictions imposed by the square root symbol (√). The expression inside the square root must be non-negative, as the square root of a negative number is undefined in the real number system.

Restrictions Imposed by the Square Root Symbol

For the expression √(x-1) to be defined, the value inside the square root, x-1, must be greater than or equal to zero. This is because the square root of a negative number is undefined in the real number system.

Solving the Inequality x-1 ≥ 0

To find the values of x that satisfy the inequality x-1 ≥ 0, we can add 1 to both sides of the inequality, resulting in x ≥ 1.

Domain of the Function p(x) = √(x-1) + 2

Based on the inequality x ≥ 1, we can conclude that the domain of the function p(x) = √(x-1) + 2 is the set of all real numbers greater than or equal to 1. This can be represented as [1, ∞).

Comparing the Domain with the Given Options

Now that we have determined the domain of the function p(x) = √(x-1) + 2, we can compare it with the given options:

A. [2, ∞) B. (-∞, 2] C. (-∞, 1] D. [1, ∞)

Selecting the Correct Answer

Based on our analysis, the correct answer is D. [1, ∞). This is because the domain of the function p(x) = √(x-1) + 2 is the set of all real numbers greater than or equal to 1.

Conclusion

In conclusion, the domain of the function p(x) = √(x-1) + 2 is the set of all real numbers greater than or equal to 1. This can be represented as [1, ∞). By understanding the restrictions imposed by the square root symbol and solving the resulting inequality, we can determine the domain of the function and select the correct answer from the given options.

Key Takeaways

The domain of a function is the set of all possible input values (x-values) for which the function is defined.
The expression inside the square root symbol must be non-negative for the function to be defined.
The inequality x ≥ 1 represents the values of x that satisfy the condition x-1 ≥ 0.
The domain of the function p(x) = √(x-1) + 2 is the set of all real numbers greater than or equal to 1, represented as [1, ∞).

Final Answer

The final answer is D. [1, ∞).

Introduction to Domain of a Function FAQs

In our previous article, we discussed the concept of the domain of a function and how to determine it for a given function. In this article, we will address some frequently asked questions related to 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 (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output.

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

A: To determine the domain of a function, you need to consider the restrictions imposed by the function. For example, if the function involves a square root, the expression inside the square root must be non-negative. If the function involves a fraction, the denominator cannot be 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 (x-values) for which the function is defined, while 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 an empty domain?

A: Yes, a function can have an empty domain. This occurs when the function is undefined for all possible input values.

Q: Can a function have a domain that is all real numbers?

A: Yes, a function can have a domain that is all real numbers. This occurs when the function is defined for all possible input values.

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

A: The domain of a function can be represented using interval notation. For example, the domain of a function can be represented as [a, b] to indicate that the function is defined for all real numbers between a and b, inclusive.

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

A: The domain of a function is significant because it determines the set of all possible input values for which the function is defined. This is important because it helps to ensure that the function is used correctly and that the results are meaningful.

Q: Can I have multiple domains for a single function?

A: No, a function can only have one domain. However, a function can have multiple intervals or sets of values that are not part of its domain.

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

A: To determine the domain of a function with multiple variables, you need to consider the restrictions imposed by each variable. For example, if the function involves a square root, the expression inside the square root must be non-negative for each variable.

Q: Can I have a function with an infinite domain?

A: Yes, a function can have an infinite domain. This occurs when the function is defined for all possible input values, including those that are not bounded by a finite interval.

Q: How do I represent an infinite domain?

A: An infinite domain can be represented using interval notation, such as (-∞, ∞) to indicate that the function is defined for all real numbers.

Conclusion

In conclusion, the domain of a function is an essential concept in mathematics that determines the set of all possible input values for which the function is defined. By understanding the domain of a function, you can ensure that the function is used correctly and that the results are meaningful.

Key Takeaways

The domain of a function is the set of all possible input values (x-values) for which the function is defined.
The domain of a function can be represented using interval notation.
The domain of a function determines the set of all possible input values for which the function is defined.
A function can have an empty domain, a finite domain, or an infinite domain.
A function can have multiple intervals or sets of values that are not part of its domain.

Final Answer

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