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

Introduction

When dealing with functions, it's essential to understand their domain, which represents the set of all possible input values for which the function is defined. In this article, we will explore the concept of a domain of all real numbers and determine which function among the given options has this property.

Understanding Domain

The domain of a function is the set of all possible input values (x-values) that can be plugged into the function without resulting in an undefined or imaginary output. In other words, it's the set of all possible x-values for which the function is defined.

Domain of All Real Numbers

A function has a domain of all real numbers if it is defined for every possible real number input. This means that the function should not have any restrictions on its input values, such as being undefined at a specific point or having a limited range of values.

Analyzing the Options

Let's analyze each of the given functions to determine which one has a domain of all real numbers.

A. $y = -2(3x)^{\frac{1}{6}}$

This function involves a fractional exponent, which can be rewritten as:

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

Since the exponent $\frac{1}{6}$ is positive, the function is defined for all real numbers, including negative values. Therefore, this function has a domain of all real numbers.

B. $y = (2x)^{\frac{1}{3}} - 7$

This function also involves a fractional exponent, which can be rewritten as:

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

Since the exponent $\frac{1}{3}$ is positive, the function is defined for all real numbers, including negative values. However, we need to consider the value of $2^{\frac{1}{3}}$, which is approximately 1.2599. This value is positive, but it's not equal to 0. If $x$ is equal to 0, the function is undefined, since it would result in division by zero. Therefore, this function does not have a domain of all real numbers.

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

This function involves a fractional exponent, which can be rewritten as:

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

Since the exponent $\frac{1}{4}$ is positive, the function is defined for all real numbers, including negative values. However, we need to consider the value of $x + 2$. If $x + 2$ is equal to 0, the function is undefined, since it would result in taking the fourth root of zero. Therefore, this function does not have a domain of all real numbers.

D. $y = -x^{\frac{1}{2}} + 5$

This function involves a square root, which can be rewritten as:

$$y = -\sqrt{x} + 5$$

Since the exponent $\frac{1}{2}$ is positive, the function is defined for all non-negative real numbers. However, if $x$ is negative, the function is undefined, since it would result in taking the square root of a negative number. Therefore, this function does not have a domain of all real numbers.

Conclusion

Based on the analysis of each function, we can conclude that only option A, $y = -2(3x)^{\frac{1}{6}}$, has a domain of all real numbers. This function is defined for every possible real number input, making it the correct answer.

Final Answer

Introduction

In our previous article, we explored the concept of a domain of all real numbers and determined which function among the given options has this property. In this article, we will provide a Q&A section to further clarify the concept and address any questions or concerns you may have.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) that can be plugged into the function without resulting in an undefined or imaginary output.

Q: What is the difference between a domain and a range?

A: The domain of a function is the set of all possible input values, while the range 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: How do I determine the domain of a function?

A: To determine the domain of a function, you need to consider the following:

• Exponents: If the exponent is positive, the function is defined for all real numbers. If the exponent is negative, the function is undefined for negative values.
• Square roots: If the expression under the square root is non-negative, the function is defined for all non-negative real numbers. If the expression under the square root is negative, the function is undefined.
• Division: If the denominator is zero, the function is undefined.
• Other restrictions: Consider any other restrictions on the input values, such as being undefined at a specific point or having a limited range of values.

Q: Can a function have a domain of all real numbers if it involves a square root?

A: Yes, a function can have a domain of all real numbers even if it involves a square root. For example, the function $y = \sqrt{x} + 5$ has a domain of all non-negative real numbers, but the function $y = \sqrt{x} - 5$ has a domain of all real numbers, since the expression under the square root is always non-negative.

Q: Can a function have a domain of all real numbers if it involves a fractional exponent?

A: Yes, a function can have a domain of all real numbers even if it involves a fractional exponent. For example, the function $y = (x + 2)^{\frac{1}{4}}$ has a domain of all real numbers, since the exponent $\frac{1}{4}$ is positive.

Q: How do I determine 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 the function is defined for every possible real number input. You can do this by:

• Checking the exponents: If the exponents are positive, the function is defined for all real numbers.
• Checking the square roots: If the expressions under the square roots are non-negative, the function is defined for all non-negative real numbers.
• Checking the division: If the denominators are non-zero, the function is defined for all real numbers.
• Checking other restrictions: Consider any other restrictions on the input values.

Conclusion

In this Q&A article, we have addressed some common questions and concerns related to the domain of all real numbers. We hope that this article has provided you with a better understanding of the concept and how to determine if a function has a domain of all real numbers.

Final Tips

• Be careful with exponents: If the exponent is positive, the function is defined for all real numbers. If the exponent is negative, the function is undefined for negative values.
• Be careful with square roots: If the expression under the square root is non-negative, the function is defined for all non-negative real numbers. If the expression under the square root is negative, the function is undefined.
• Be careful with division: If the denominator is zero, the function is undefined.
• Check other restrictions: Consider any other restrictions on the input values.