Select The Correct Answer.Which Function Has A Domain Of All Real Numbers?A. $y = -x^{\frac{1}{2}} + 5$ B. $y = -2(3x)^{\frac{1}{6}}$ C. $y = (x+2)^{\frac{1}{4}}$ D. $y = (2x)^{\frac{1}{3}} - 7$

When dealing with functions, understanding the domain is crucial. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this article, we will explore which function has a domain of all real numbers.

Understanding the Domain of a Function

The domain of a function is determined by the values of x that make the function undefined or imaginary. For example, in the function $y = \frac{1}{x}$, the domain is all real numbers except for x = 0, because division by zero is undefined.

Analyzing the Options

Let's analyze each option to determine which function has a domain of all real numbers.

Option A: $y = -x^{\frac{1}{2}} + 5$

The function $y = -x^{\frac{1}{2}} + 5$ involves a square root. The square root of a negative number is undefined in the real number system. Therefore, the domain of this function is all non-negative real numbers, not all real numbers.

Option B: $y = -2(3x)^{\frac{1}{6}}$

This function involves a sixth root. The sixth root of a negative number is defined in the real number system, but the function is still restricted to real numbers. However, we need to consider the inner expression $(3x)^{\frac{1}{6}}$. Since the exponent is $\frac{1}{6}$, the function is defined for all real numbers, but we need to check if there are any restrictions on the inner expression.

Option C: $y = (x+2)^{\frac{1}{4}}$

This function involves a fourth root. Similar to the previous option, the fourth root of a negative number is defined in the real number system. However, we need to consider the inner expression $(x+2)^{\frac{1}{4}}$. Since the exponent is $\frac{1}{4}$, the function is defined for all real numbers, but we need to check if there are any restrictions on the inner expression.

Option D: $y = (2x)^{\frac{1}{3}} - 7$

This function involves a cube root. The cube root of a negative number is defined in the real number system. However, we need to consider the inner expression $(2x)^{\frac{1}{3}}$. Since the exponent is $\frac{1}{3}$, the function is defined for all real numbers, but we need to check if there are any restrictions on the inner expression.

Determining the Correct Answer

To determine which function has a domain of all real numbers, we need to analyze the inner expressions of each function. In options B, C, and D, the inner expressions involve a root with an even exponent. Since the exponent is even, the function is defined for all real numbers, but we need to check if there are any restrictions on the inner expression.

However, in option B, the inner expression is $(3x)^{\frac{1}{6}}$. Since the exponent is $\frac{1}{6}$, which is not an even number, the function is not defined for all real numbers. In options C and D, the inner expressions are $(x+2)^{\frac{1}{4}}$ and $(2x)^{\frac{1}{3}}$, respectively. Since the exponents are $\frac{1}{4}$ and $\frac{1}{3}$, which are not even numbers, the functions are not defined for all real numbers.

However, we can see that option B has a restriction on the inner expression, but it is not a restriction on the domain of the function. The function is defined for all real numbers, but the inner expression is not defined for all real numbers.

Conclusion

After analyzing each option, we can conclude that option B, $y = -2(3x)^{\frac{1}{6}}$, has a domain of all real numbers. The function is defined for all real numbers, but the inner expression is not defined for all real numbers.

Final Answer

In the previous article, we discussed the concept of the domain of a function and analyzed four different functions to determine which one has a domain of all real numbers. 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 (x-values) for which the function is defined.

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 a function involves a square root, the input value must be non-negative. If a function involves a fraction, the denominator cannot be zero.

Q: What are some common restrictions on the domain of a function?

A: Some common restrictions on the domain of a function include:

• Division by zero: The denominator of a fraction cannot be zero.
• Square root of a negative number: The input value must be non-negative.
• Logarithm of a non-positive number: The input value must be positive.
• Exponent with a negative exponent: The input value must be positive.

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 $y = x^2$ has a domain of all real numbers.

Q: How do I know if a function has a domain of all real numbers?

A: To determine if a function has a domain of all real numbers, you need to check if there are any restrictions on the input values. If there are no restrictions, then the function has a domain of all real numbers.

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 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 $y = \sqrt{x}$ has a domain of all non-negative real numbers.

Q: How do I graph a function with a domain that is a subset of the real numbers?

A: To graph a function with a domain that is a subset of the real numbers, you need to identify the subset of the real numbers that is the domain of the function. Then, you can graph the function on that subset of the real numbers.

Q: Can a function have a domain that is empty?

A: Yes, a function can have a domain that is empty. For example, the function $y = \frac{1}{x}$ has a domain that is empty, because the denominator cannot be zero.

Q: How do I determine if a function has a domain that is empty?

A: To determine if a function has a domain that is empty, you need to check if there are any restrictions on the input values. If there are no restrictions, then the function has a domain that is not empty. If there are restrictions, then the function may have a domain that is empty.

Conclusion

In this article, we answered some frequently asked questions about the domain of a function. We discussed how to determine the domain of a function, common restrictions on the domain, and how to graph a function with a domain that is a subset of the real numbers. We also discussed how to determine if a function has a domain that is empty.