Determine the validity of a function

Oct 13, 2025 by miodeya.com 8 views

In mathematics, the domain of a function refers to the set of all possible input values for which the function is defined. On the other hand, the range of a function is the set of all possible output values. In this article, we will explore which of the given functions has a limited domain.

What is a Limited Domain?

A limited domain is a domain that is not the entire set of real numbers. In other words, it is a domain that is restricted to a specific subset of real numbers. For example, the function $f(x) = \sqrt{x}$ has a limited domain because the square root of a negative number is not defined in the real number system.

Option #1: $f(x) = |x|$

The absolute value function $f(x) = |x|$ is defined for all real numbers. This means that the domain of the function is the entire set of real numbers, from negative infinity to positive infinity. Therefore, the absolute value function does not have a limited domain.

Option #2: $f(x) = x^2$

The quadratic function $f(x) = x^2$ is also defined for all real numbers. This means that the domain of the function is the entire set of real numbers, from negative infinity to positive infinity. Therefore, the quadratic function does not have a limited domain.

Option #3: $f(x) = \sqrt{x}$

The square root function $f(x) = \sqrt{x}$ is defined only for non-negative real numbers. This means that the domain of the function is the set of all non-negative real numbers, from 0 to positive infinity. Therefore, the square root function has a limited domain.

Conclusion

Based on the analysis above, we can conclude that the function with a limited domain is the square root function $f(x) = \sqrt{x}$ . This is because the square root function is defined only for non-negative real numbers, whereas the absolute value and quadratic functions are defined for all real numbers.

Why is Understanding Domain and Range Important?

Understanding the domain and range of a function is crucial in mathematics and other fields. It helps us to:

Avoid undefined values: By knowing the domain of a function, we can avoid using input values that would result in undefined or imaginary numbers.
Determine the validity of a function: The domain of a function determines whether the function is valid or not. If the domain is not specified, we cannot determine the validity of the function.
Solve equations and inequalities: Understanding the domain and range of a function is essential for solving equations and inequalities involving the function.
Graph functions: The domain and range of a function determine the shape and behavior of its graph.

Real-World Applications of Domain and Range

The concept of domain and range has numerous real-world applications in:

Physics and Engineering: In physics and engineering, the domain and range of functions are used to model real-world phenomena, such as motion, energy, and forces.
Computer Science: In computer science, the domain and range of functions are used to develop algorithms and data structures, such as sorting and searching.
Economics: In economics, the domain and range of functions are used to model economic systems, such as supply and demand.

Common Functions with Limited Domains

Some common functions with limited domains include:

Square root function: $f(x) = \sqrt{x}$ , defined only for non-negative real numbers.
Reciprocal function: $f(x) = \frac{1}{x}$ , defined only for non-zero real numbers.
Exponential function: $f(x) = e^x$ , defined only for real numbers.

Conclusion

In this article, we will answer some frequently asked questions about domain and range.

Q: What is the difference between domain and range?

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.

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

A: Knowing the domain of a function is important because it helps us to avoid undefined values, determine the validity of a function, solve equations and inequalities, and graph functions.

Q: What is a limited domain?

A: A limited domain is a domain that is not the entire set of real numbers. In other words, it is a domain that is restricted to a specific subset of real numbers.

Q: Which of the following functions has a limited domain?

A: The function $f(x) = \sqrt{x}$ has a limited domain because the square root of a negative number is not defined in the real number system.

Q: What is the domain of the function $f(x) = |x|$ ?

A: The domain of the function $f(x) = |x|$ is the entire set of real numbers, from negative infinity to positive infinity.

Q: What is the range of the function $f(x) = x^2$ ?

A: The range of the function $f(x) = x^2$ is the set of all non-negative real numbers, from 0 to positive infinity.

Q: Can a function have a limited range?

A: Yes, a function can have a limited range. For example, the function $f(x) = \sin(x)$ has a limited range because the sine function oscillates between -1 and 1.

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

A: To determine the domain and range of a function, you need to analyze the function and identify any restrictions on the input values and output values.

Q: What are some common functions with limited domains?

A: Some common functions with limited domains include:

Square root function: $f(x) = \sqrt{x}$ , defined only for non-negative real numbers.
Reciprocal function: $f(x) = \frac{1}{x}$ , defined only for non-zero real numbers.
Exponential function: $f(x) = e^x$ , defined only for real numbers.

Q: Can a function have a domain that is a subset of the real numbers?

A: Yes, a function can have a domain that is a subset of the real numbers. For example, the function $f(x) = \sqrt{x}$ has a domain that is a subset of the real numbers, specifically the set of non-negative real numbers.

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

A: To graph a function with a limited domain, you need to identify the domain and range of the function and then plot the function on a coordinate plane.

Q: Can a function have a range that is a subset of the real numbers?

A: Yes, a function can have a range that is a subset of the real numbers. For example, the function $f(x) = \sin(x)$ has a range that is a subset of the real numbers, specifically the set of real numbers between -1 and 1.

Conclusion

In conclusion, understanding the domain and range of a function is crucial in mathematics and other fields. By knowing the domain and range of a function, we can avoid undefined values, determine the validity of a function, solve equations and inequalities, and graph functions.