Which Statement Best Describes The Domain And Range Of $f(x)=-(7)^x$ And $g(x)=7^x$?A. $f(x$\] And $g(x$\] Have The Same Domain And The Same Range.B. $f(x$\] And $g(x$\] Have The Same Domain But

Introduction

In mathematics, the domain and range of a function are crucial concepts that help us understand the behavior and characteristics of the function. The domain of a function is the set of all possible input values (x-values) that the function can accept, while the range is the set of all possible output values (y-values) that the function can produce. In this article, we will explore the domain and range of two exponential functions, $f(x) = -(7)^x$ and $g(x) = 7^x$.

Domain and Range of Exponential Functions

Exponential functions are of the form $f(x) = a^x$, where $a$ is a positive real number. The domain of an exponential function is all real numbers, and the range is also all real numbers, except for the case where $a = 0$. In this case, the range is only positive real numbers.

Domain of $f(x) = -(7)^x$ and $g(x) = 7^x$

The domain of both $f(x) = -(7)^x$ and $g(x) = 7^x$ is all real numbers. This is because the base of the exponential function, 7, is a positive real number, and the exponent, x, can be any real number.

Range of $f(x) = -(7)^x$ and $g(x) = 7^x$

The range of $f(x) = -(7)^x$ is all negative real numbers, while the range of $g(x) = 7^x$ is all positive real numbers. This is because the negative sign in front of the exponential function in $f(x)$ causes the output to be negative, while the absence of a negative sign in $g(x)$ causes the output to be positive.

Comparison of Domain and Range

Since the domain of both $f(x) = -(7)^x$ and $g(x) = 7^x$ is all real numbers, they have the same domain. However, the range of $f(x) = -(7)^x$ is all negative real numbers, while the range of $g(x) = 7^x$ is all positive real numbers. Therefore, they do not have the same range.

Conclusion

In conclusion, the domain of both $f(x) = -(7)^x$ and $g(x) = 7^x$ is all real numbers, but the range of $f(x) = -(7)^x$ is all negative real numbers, while the range of $g(x) = 7^x$ is all positive real numbers. Therefore, they have the same domain but not the same range.

Answer to the Question

The correct answer to the question is:

B. $f(x)$ and $g(x)$ have the same domain but not the same range.

Final Thoughts

Q: What is the domain of an exponential function?

A: The domain of an exponential function is all real numbers. This means that the exponent, x, can be any real number, and the function will still be defined.

Q: What is the range of an exponential function?

A: The range of an exponential function is all positive real numbers, except for the case where the base is 0. In this case, the range is only positive real numbers.

Q: How does the base of an exponential function affect its domain and range?

A: The base of an exponential function does not affect its domain, but it does affect its range. If the base is a positive real number, the range is all positive real numbers. If the base is a negative real number, the range is all negative real numbers.

Q: How does the exponent of an exponential function affect its domain and range?

A: The exponent of an exponential function does not affect its range, but it does affect its domain. If the exponent is a real number, the function is defined for all real numbers. If the exponent is not a real number, the function is not defined.

Q: What is the difference between the domain and range of $f(x) = -(7)^x$ and $g(x) = 7^x$?

A: The domain of both $f(x) = -(7)^x$ and $g(x) = 7^x$ is all real numbers. However, the range of $f(x) = -(7)^x$ is all negative real numbers, while the range of $g(x) = 7^x$ is all positive real numbers.

Q: Can the domain and range of an exponential function be the same?

A: No, the domain and range of an exponential function cannot be the same. The domain is all real numbers, while the range is either all positive real numbers or all negative real numbers.

Q: How can I determine the domain and range of an exponential function?

A: To determine the domain and range of an exponential function, you can use the following steps:

1. Identify the base of the exponential function.
2. Determine if the base is a positive or negative real number.
3. If the base is a positive real number, the range is all positive real numbers.
4. If the base is a negative real number, the range is all negative real numbers.
5. The domain is always all real numbers.

Q: What are some common mistakes to avoid when working with exponential functions?

A: Some common mistakes to avoid when working with exponential functions include:

1. Assuming that the domain and range of an exponential function are the same.
2. Failing to consider the base of the exponential function when determining its range.
3. Not recognizing that the exponent of an exponential function can affect its domain.

Q: How can I apply my understanding of domain and range to real-world problems?

A: You can apply your understanding of domain and range to real-world problems in a variety of ways, such as:

1. Modeling population growth or decline using exponential functions.
2. Analyzing the behavior of financial investments or loans using exponential functions.
3. Understanding the behavior of chemical reactions or physical systems using exponential functions.

By understanding the domain and range of exponential functions, you can better analyze and interpret the behavior of these functions in various fields, such as science, engineering, and economics.