Consider The Functions F ( X ) = ( 4 5 ) X F(x)=\left(\frac{4}{5}\right)^x F ( X ) = ( 5 4 ​ ) X And G ( X ) = ( 4 5 ) X + 6 G(x)=\left(\frac{4}{5}\right)^x+6 G ( X ) = ( 5 4 ​ ) X + 6 . What Are The Ranges Of The Two Functions?$ \begin{array}{l} f(x):{y \mid Y\ \textgreater \ 0} \ g(x):{y \mid Y\ \textgreater \

Introduction

In mathematics, functions are used to describe the relationship between variables. Exponential functions, in particular, are used to describe growth or decay. In this article, we will explore the ranges of two exponential functions, $f(x)=\left(\frac{4}{5}\right)^x$ and $g(x)=\left(\frac{4}{5}\right)^x+6$. Understanding the ranges of these functions is crucial in various mathematical and real-world applications.

The Function $f(x)=\left(\frac{4}{5}\right)^x$

The function $f(x)=\left(\frac{4}{5}\right)^x$ is an exponential function with base $\frac{4}{5}$. This function is defined for all real numbers $x$. To find the range of this function, we need to consider the possible values of $y$.

Since the base of the function is $\frac{4}{5}$, which is less than 1, the function is decreasing as $x$ increases. This means that as $x$ gets larger, the value of $f(x)$ gets smaller. On the other hand, as $x$ gets smaller, the value of $f(x)$ gets larger.

To find the range of this function, we can consider the following:

• As $x$ approaches negative infinity, $f(x)$ approaches infinity.
• As $x$ approaches positive infinity, $f(x)$ approaches 0.
• For any real number $x$, $f(x)$ is always positive.

Therefore, the range of the function $f(x)=\left(\frac{4}{5}\right)^x$ is $\{y \mid y > 0\}$.

The Function $g(x)=\left(\frac{4}{5}\right)^x+6$

The function $g(x)=\left(\frac{4}{5}\right)^x+6$ is also an exponential function with base $\frac{4}{5}$. However, this function has an additional constant term of 6. To find the range of this function, we need to consider the possible values of $y$.

Since the base of the function is $\frac{4}{5}$, which is less than 1, the function is decreasing as $x$ increases. This means that as $x$ gets larger, the value of $g(x)$ gets smaller. On the other hand, as $x$ gets smaller, the value of $g(x)$ gets larger.

To find the range of this function, we can consider the following:

• As $x$ approaches negative infinity, $g(x)$ approaches infinity.
• As $x$ approaches positive infinity, $g(x)$ approaches 6.
• For any real number $x$, $g(x)$ is always greater than 6.

Therefore, the range of the function $g(x)=\left(\frac{4}{5}\right)^x+6$ is $\{y \mid y > 6\}$.

Comparison of the Ranges

In summary, the range of the function $f(x)=\left(\frac{4}{5}\right)^x$ is $\{y \mid y > 0\}$, while the range of the function $g(x)=\left(\frac{4}{5}\right)^x+6$ is $\{y \mid y > 6\}$. This means that the function $g(x)$ has a minimum value of 6, while the function $f(x)$ has no minimum value.

Conclusion

In conclusion, the ranges of the two exponential functions $f(x)=\left(\frac{4}{5}\right)^x$ and $g(x)=\left(\frac{4}{5}\right)^x+6$ are $\{y \mid y > 0\}$ and $\{y \mid y > 6\}$, respectively. Understanding the ranges of these functions is crucial in various mathematical and real-world applications.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Algebra, 2nd edition, Michael Artin
• [3] Exponential Functions, Wolfram MathWorld

Discussion

Introduction

In our previous article, we explored the ranges of two exponential functions, $f(x)=\left(\frac{4}{5}\right)^x$ and $g(x)=\left(\frac{4}{5}\right)^x+6$. In this article, we will answer some frequently asked questions about exponential functions and their ranges.

Q: What is an exponential function?

A: An exponential function is a function of the form $f(x) = a^x$, where $a$ is a positive real number. Exponential functions are used to describe growth or decay.

Q: What is the range of an exponential function?

A: The range of an exponential function depends on the base of the function. If the base is greater than 1, the range is all positive real numbers. If the base is between 0 and 1, the range is all positive real numbers. If the base is 1, the range is all real numbers.

Q: How do you find the range of an exponential function?

A: To find the range of an exponential function, you need to consider the possible values of $y$. You can do this by analyzing the behavior of the function as $x$ approaches positive or negative infinity.

Q: What is the difference between the ranges of $f(x)=\left(\frac{4}{5}\right)^x$ and $g(x)=\left(\frac{4}{5}\right)^x+6$?

A: The range of $f(x)=\left(\frac{4}{5}\right)^x$ is $\{y \mid y > 0\}$, while the range of $g(x)=\left(\frac{4}{5}\right)^x+6$ is $\{y \mid y > 6\}$. This means that the function $g(x)$ has a minimum value of 6, while the function $f(x)$ has no minimum value.

Q: What are some real-world applications of exponential functions?

A: Exponential functions have many real-world applications, including:

• Population growth: Exponential functions can be used to model population growth.
• Chemical reactions: Exponential functions can be used to model chemical reactions.
• Finance: Exponential functions can be used to model interest rates and investments.
• Biology: Exponential functions can be used to model the growth of bacteria and other organisms.

Q: How do you graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer program. You can also use a table of values to create a graph.

Q: What is the domain of an exponential function?

A: The domain of an exponential function is all real numbers.

Q: What is the difference between an exponential function and a logarithmic function?

A: An exponential function is a function of the form $f(x) = a^x$, while a logarithmic function is a function of the form $f(x) = \log_a(x)$. Exponential functions describe growth or decay, while logarithmic functions describe the inverse of exponential functions.

Conclusion

In conclusion, exponential functions have many real-world applications and are used to describe growth or decay. Understanding the ranges of exponential functions is crucial in various mathematical and real-world applications.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Algebra, 2nd edition, Michael Artin
• [3] Exponential Functions, Wolfram MathWorld

Discussion

What are some other real-world applications of exponential functions? How do you think the ranges of these functions affect their applications? Share your thoughts in the comments below!