A Function Of The Form F ( X ) = A B B X F(x) = A B B^x F ( X ) = Ab B X Is Modified So That The B B B Value Remains The Same, But The A A A Value Is Increased By 2. How Do The Domain And Range Of The New Function Compare To The Domain And Range Of The Original

Introduction

In mathematics, functions are used to describe the relationship between variables. A function of the form $f(x) = a b b^x$ is a type of exponential function, where $a$ is the coefficient, $b$ is the base, and $x$ is the variable. In this article, we will explore how the domain and range of this function change when the $a$ value is increased by 2, while the $b$ value remains the same.

Original Function

The original function is given by $f(x) = a b b^x$. To understand the domain and range of this function, we need to consider the properties of exponential functions.

• Domain: The domain of an exponential function is all real numbers, unless the base is negative, in which case the domain is restricted to non-negative real numbers. In this case, since $b$ is the base, the domain of the original function is all real numbers, $(-\infty, \infty)$.
• Range: The range of an exponential function is all positive real numbers, unless the base is negative, in which case the range is restricted to non-positive real numbers. In this case, since $b$ is the base, the range of the original function is all positive real numbers, $(0, \infty)$.

Modified Function

The modified function is obtained by increasing the $a$ value by 2, while keeping the $b$ value the same. The new function is given by $f(x) = (a + 2) b b^x$.

Domain Comparison

To compare the domain of the modified function with the domain of the original function, we need to consider the properties of the new function.

• Domain: Since the $a$ value has been increased by 2, the coefficient of the new function is larger than the coefficient of the original function. However, the base $b$ remains the same, and the domain of the modified function is still all real numbers, $(-\infty, \infty)$.
• Conclusion: The domain of the modified function is the same as the domain of the original function.

Range Comparison

To compare the range of the modified function with the range of the original function, we need to consider the properties of the new function.

• Range: Since the $a$ value has been increased by 2, the coefficient of the new function is larger than the coefficient of the original function. This means that the new function will produce larger values than the original function for the same input.
• Conclusion: The range of the modified function is the same as the range of the original function, but it is shifted upwards.

Graphical Representation

To visualize the comparison between the original and modified functions, we can plot their graphs.

• Original Function: The graph of the original function is a curve that passes through the point $(0, a)$ and has a horizontal asymptote at $y = 0$.
• Modified Function: The graph of the modified function is a curve that passes through the point $(0, a + 2)$ and has a horizontal asymptote at $y = 0$.

Conclusion

In conclusion, when the $a$ value is increased by 2 in the function $f(x) = a b b^x$, the domain and range of the new function remain the same as the domain and range of the original function. However, the new function produces larger values than the original function for the same input.

Key Takeaways

• The domain of the modified function is the same as the domain of the original function.
• The range of the modified function is the same as the range of the original function, but it is shifted upwards.
• The new function produces larger values than the original function for the same input.

Final Thoughts

Introduction

In our previous article, we explored how the domain and range of the function $f(x) = a b b^x$ change when the $a$ value is increased by 2, while the $b$ value remains the same. In this article, we will answer some frequently asked questions (FAQs) related to this topic.

Q&A

Q: What is the domain of the modified function?

A: The domain of the modified function is the same as the domain of the original function, which is all real numbers, $(-\infty, \infty)$.

Q: What is the range of the modified function?

A: The range of the modified function is the same as the range of the original function, which is all positive real numbers, $(0, \infty)$. However, the new function produces larger values than the original function for the same input.

Q: How does the modified function compare to the original function in terms of its graph?

A: The graph of the modified function is a curve that passes through the point $(0, a + 2)$ and has a horizontal asymptote at $y = 0$. This means that the modified function has the same shape as the original function, but it is shifted upwards.

Q: What are the implications of the modified function for the application of this function in various fields?

A: The modified function has important implications for the application of this function in various fields, such as economics, finance, and science. For example, in economics, the modified function can be used to model the growth of a population or the demand for a product. In finance, the modified function can be used to model the growth of an investment or the return on investment. In science, the modified function can be used to model the growth of a chemical reaction or the decay of a radioactive substance.

Q: Can the modified function be used to model any real-world phenomenon?

A: Yes, the modified function can be used to model any real-world phenomenon that exhibits exponential growth or decay. For example, the modified function can be used to model the growth of a population, the decay of a radioactive substance, or the growth of a chemical reaction.

Q: How can the modified function be used in real-world applications?

A: The modified function can be used in real-world applications such as:

• Modeling the growth of a population or the demand for a product in economics.
• Modeling the growth of an investment or the return on investment in finance.
• Modeling the growth of a chemical reaction or the decay of a radioactive substance in science.
• Modeling the growth of a company or the return on investment in business.

Q: What are the limitations of the modified function?

A: The modified function has some limitations, such as:

• It assumes that the growth or decay is exponential, which may not always be the case in real-world phenomena.
• It assumes that the base $b$ remains the same, which may not always be the case in real-world phenomena.
• It assumes that the coefficient $a$ remains the same, which may not always be the case in real-world phenomena.

Q: Can the modified function be used to model any non-exponential growth or decay?

A: No, the modified function can only be used to model exponential growth or decay. If the growth or decay is not exponential, a different function should be used.

Conclusion

In conclusion, the modified function $f(x) = (a + 2) b b^x$ has the same domain and range as the original function $f(x) = a b b^x$, but it produces larger values than the original function for the same input. The modified function has important implications for the application of this function in various fields, such as economics, finance, and science. However, it has some limitations, such as assuming that the growth or decay is exponential, and assuming that the base $b$ and coefficient $a$ remain the same.

Key Takeaways

• The domain of the modified function is the same as the domain of the original function.
• The range of the modified function is the same as the range of the original function, but it is shifted upwards.
• The modified function produces larger values than the original function for the same input.
• The modified function has important implications for the application of this function in various fields.
• The modified function has some limitations, such as assuming that the growth or decay is exponential, and assuming that the base $b$ and coefficient $a$ remain the same.