What Are The Domain And Range Of The Function $f(x) = 4(\sqrt[3]{81})^x$?A. \[$\{x \mid X \text{ Is A Real Number} \} : \{y \mid Y \ \textgreater \ 0\}\$\]B. \[$\{x \mid X \ \textgreater \ 4\} : \{y \mid Y \ \textgreater \

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Introduction

In mathematics, functions are used to describe the relationship between two variables. 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 this article, we will explore the domain and range of the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x.

What is the Domain of the Function?

The domain of a function is the set of all possible input values. In other words, it is the set of all values of xx for which the function is defined. To find the domain of the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x, we need to consider the restrictions on the input values.

The function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x is defined for all real numbers xx. This is because the cube root of 81 is a real number, and raising a real number to any power results in a real number. Therefore, the domain of the function is the set of all real numbers.

What is the Range of the Function?

The range of a function is the set of all possible output values. In other words, it is the set of all values of yy that the function can produce. To find the range of the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x, we need to consider the behavior of the function as xx varies.

The function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x is an exponential function with base 813\sqrt[3]{81}. Since 813=3\sqrt[3]{81} = 3, the function can be rewritten as f(x)=4(3)xf(x) = 4(3)^x. This function is increasing for all values of xx, and it has a minimum value of 0 when x=−∞x = -\infty. As xx increases, the function grows exponentially, and it has no maximum value.

Finding the Range of the Function

To find the range of the function, we need to consider the behavior of the function as xx varies. Since the function is increasing for all values of xx, we can conclude that the range of the function is the set of all positive real numbers.

Conclusion

In conclusion, the domain of the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x is the set of all real numbers, while the range is the set of all positive real numbers. This is because the function is defined for all real numbers, and it produces only positive output values.

Answer

The correct answer is:

A. {x∣x is a real number}:{y∣y \textgreater 0}\{x \mid x \text{ is a real number} \} : \{y \mid y \ \textgreater \ 0\}

Q: What is the domain of the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x?

A: The domain of the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x is the set of all real numbers. This is because the cube root of 81 is a real number, and raising a real number to any power results in a real number.

Q: Why is the domain of the function all real numbers?

A: The domain of the function is all real numbers because there are no restrictions on the input values. The function is defined for all real numbers, and it produces a real number output for every real number input.

Q: What is the range of the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x?

A: The range of the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x is the set of all positive real numbers. This is because the function is an exponential function with base 813\sqrt[3]{81}, and it produces only positive output values.

Q: Why is the range of the function all positive real numbers?

A: The range of the function is all positive real numbers because the function is increasing for all values of xx. As xx increases, the function grows exponentially, and it has no maximum value. Therefore, the function produces only positive output values.

Q: Can the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x produce zero or negative output values?

A: No, the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x cannot produce zero or negative output values. This is because the function is an exponential function with base 813\sqrt[3]{81}, and it produces only positive output values.

Q: What is the minimum value of the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x?

A: The minimum value of the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x is 0, which occurs when x=−∞x = -\infty. As xx increases, the function grows exponentially, and it has no maximum value.

Q: Can the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x be defined for complex numbers?

A: No, the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x is not defined for complex numbers. This is because the cube root of 81 is a real number, and raising a real number to any power results in a real number.

Q: What is the significance of the domain and range of the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x?

A: The domain and range of the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x are significant because they describe the behavior of the function. The domain indicates the set of all possible input values, while the range indicates the set of all possible output values.

Conclusion

In conclusion, the domain and range of the function f(x)=4(813)xf(x) = 4(\sqrt[3]{81})^x are the set of all real numbers and the set of all positive real numbers, respectively. This is because the function is defined for all real numbers, and it produces only positive output values.