A Function Of The Form F ( X ) = A B X F(x) = A B^x F ( X ) = A 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

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Introduction

In mathematics, functions of the form $f(x) = a b^x$ are commonly used to model exponential growth and decay. These functions have a wide range of applications in various fields, including physics, engineering, and economics. In this article, we will discuss how the domain and range of a function of the form $f(x) = a b^x$ change when the $a$ value is increased by 2, while the $b$ value remains the same.

Original Function: Domain and Range

The original function is of the form $f(x) = a b^x$. The domain of this function is all real numbers, denoted by $(-\infty, \infty)$. This is because the function is defined for all values of $x$.

The range of the function depends on the value of $a$. If $a$ is positive, the range is $(0, \infty)$. If $a$ is negative, the range is $(-\infty, 0)$. If $a$ is zero, the function is a constant function, and the range is $\{0\}$.

Modified Function: Domain and Range

When the $a$ value is increased by 2, the new function becomes $f(x) = (a + 2) b^x$. The domain of this function remains the same as the original function, which is all real numbers, denoted by $(-\infty, \infty)$.

The range of the modified function depends on the value of $a$. If $a$ is positive, the range is $(2a, \infty)$. If $a$ is negative, the range is $(-\infty, 2a)$. If $a$ is zero, the function is a constant function, and the range is $\{2a\}$.

Comparison of Domain and Range

Comparing the domain and range of the original and modified functions, we can see that the domain remains the same, which is all real numbers. However, the range changes depending on the value of $a$.

If $a$ is positive, the range of the modified function is $(2a, \infty)$, which is a subset of the range of the original function $(0, \infty)$. This means that the modified function has a smaller range than the original function.

If $a$ is negative, the range of the modified function is $(-\infty, 2a)$, which is a subset of the range of the original function $(-\infty, 0)$. This means that the modified function has a smaller range than the original function.

If $a$ is zero, the range of the modified function is $\{2a\}$, which is a subset of the range of the original function $\{0\}$. This means that the modified function has a smaller range than the original function.

Conclusion

In conclusion, when the $a$ value is increased by 2 in a function of the form $f(x) = a b^x$, the domain remains the same, which is all real numbers. However, the range changes depending on the value of $a$. If $a$ is positive, the range of the modified function is a subset of the range of the original function. If $a$ is negative, the range of the modified function is a subset of the range of the original function. If $a$ is zero, the range of the modified function is a subset of the range of the original function.

Example

Let's consider an example to illustrate the concept. Suppose we have a function $f(x) = 2 b^x$, where $b$ is a positive constant. The domain of this function is all real numbers, and the range is $(0, \infty)$.

If we increase the $a$ value by 2, the new function becomes $f(x) = 4 b^x$. The domain of this function remains the same, which is all real numbers. However, the range changes to $(8, \infty)$.

Graphical Representation

The graph of the original function $f(x) = 2 b^x$ is a curve that passes through the origin and extends to infinity in the positive direction. The graph of the modified function $f(x) = 4 b^x$ is a curve that passes through the origin and extends to infinity in the positive direction, but it is steeper than the graph of the original function.

Real-World Applications

Functions of the form $f(x) = a b^x$ have many real-world applications, including modeling population growth, chemical reactions, and financial investments. In these applications, the value of $a$ can change over time, and the domain and range of the function can change accordingly.

Limitations

One limitation of this article is that it assumes that the value of $b$ remains the same. In reality, the value of $b$ can change over time, and this can affect the domain and range of the function.

Future Research

Future research can focus on exploring the effects of changing the value of $b$ on the domain and range of the function. This can provide a more comprehensive understanding of the behavior of functions of the form $f(x) = a b^x$.

Conclusion

In conclusion, when the $a$ value is increased by 2 in a function of the form $f(x) = a b^x$, the domain remains the same, which is all real numbers. However, the range changes depending on the value of $a$. If $a$ is positive, the range of the modified function is a subset of the range of the original function. If $a$ is negative, the range of the modified function is a subset of the range of the original function. If $a$ is zero, the range of the modified function is a subset of the range of the original function.

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Introduction

In our previous article, we discussed how the domain and range of a function of the form $f(x) = a b^x$ 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 related to this topic.

Q: What is the domain of the modified function?

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

Q: How does the range of the modified function compare to the range of the original function?

A: The range of the modified function depends on the value of $a$. If $a$ is positive, the range of the modified function is $(2a, \infty)$, which is a subset of the range of the original function $(0, \infty)$. If $a$ is negative, the range of the modified function is $(-\infty, 2a)$, which is a subset of the range of the original function $(-\infty, 0)$. If $a$ is zero, the range of the modified function is $\{2a\}$, which is a subset of the range of the original function $\{0\}$.

Q: What happens to the graph of the modified function compared to the graph of the original function?

A: The graph of the modified function is a curve that passes through the origin and extends to infinity in the positive direction, but it is steeper than the graph of the original function.

Q: Can the value of $b$ change over time?

A: Yes, the value of $b$ can change over time, and this can affect the domain and range of the function.

Q: How does the value of $a$ affect the range of the modified function?

A: The value of $a$ affects the range of the modified function by shifting the range up or down. If $a$ is positive, the range of the modified function is shifted up by $2a$. If $a$ is negative, the range of the modified function is shifted down by $2a$.

Q: Can the value of $a$ be negative?

A: Yes, the value of $a$ can be negative, and this can affect the range of the modified function.

Q: What is the significance of the value of $a$ in the function $f(x) = a b^x$?

A: The value of $a$ in the function $f(x) = a b^x$ represents the vertical stretch or compression of the graph of the function. A positive value of $a$ represents a vertical stretch, while a negative value of $a$ represents a vertical compression.

Q: Can the value of $b$ be negative?

A: No, the value of $b$ cannot be negative, as this would result in a function that is not defined for all real numbers.

Q: How does the value of $b$ affect the domain and range of the function?

A: The value of $b$ affects the domain and range of the function by changing the rate of growth or decay of the function. A value of $b$ greater than 1 represents exponential growth, while a value of $b$ less than 1 represents exponential decay.

Q: Can the value of $b$ be 1?

A: Yes, the value of $b$ can be 1, and this results in a function that is a constant function.

Conclusion

In conclusion, the domain and range of a function of the form $f(x) = a b^x$ change when the $a$ value is increased by 2, while the $b$ value remains the same. The domain remains the same, which is all real numbers, but the range changes depending on the value of $a$. We hope that this Q&A article has provided a better understanding of this topic.