Select The Correct Answer.Which Function Is The Inverse Of $f(x)=50,000(0.8)^x$?A. $f^{-1}(x)=\log_{0.8}\left(\frac{x}{20,000}\right)$B. \$f^{-1}(x)=\log_{50,000}\left(\frac{x}{0.8}\right)$[/tex\]C.

Introduction

Inverse functions are a fundamental concept in mathematics, particularly in algebra and calculus. They play a crucial role in solving equations, modeling real-world phenomena, and understanding the behavior of functions. In this article, we will delve into the concept of inverse functions, explore the given function $f(x)=50,000(0.8)^x$, and determine the correct inverse function among the provided options.

Understanding Inverse Functions

An inverse function is a function that reverses the operation of another function. In other words, if we have a function $f(x)$, its inverse function $f^{-1}(x)$ will take the output of $f(x)$ and return the original input. The inverse function is denoted by $f^{-1}(x)$ and is read as "f inverse of x".

To find the inverse of a function, we need to follow these steps:

1. Replace $f(x)$ with $y$.
2. Swap the roles of $x$ and $y$.
3. Solve for $y$.

The Given Function

The given function is $f(x)=50,000(0.8)^x$. To find its inverse, we will follow the steps outlined above.

Step 1: Replace $f(x)$ with $y$

$$y = 50,000(0.8)^x$$

Step 2: Swap the roles of $x$ and $y$

$$x = 50,000(0.8)^y$$

Step 3: Solve for $y$

To solve for $y$, we need to isolate it on one side of the equation. We can do this by using logarithms.

$$\log_{0.8}(x) = \log_{0.8}(50,000(0.8)^y)$$

Using the property of logarithms that $\log_a(a^b) = b$, we can simplify the equation:

$$\log_{0.8}(x) = y\log_{0.8}(0.8) + \log_{0.8}(50,000)$$

Since $\log_{0.8}(0.8) = 1$, we can further simplify the equation:

$$\log_{0.8}(x) = y + \log_{0.8}(50,000)$$

Now, we can isolate $y$ by subtracting $\log_{0.8}(50,000)$ from both sides:

$$y = \log_{0.8}(x) - \log_{0.8}(50,000)$$

Using the property of logarithms that $\log_a(b) - \log_a(c) = \log_a(\frac{b}{c})$, we can simplify the equation:

$$y = \log_{0.8}\left(\frac{x}{50,000}\right)$$

However, we need to express the equation in terms of $\log_{50,000}$, not $\log_{0.8}$. We can do this by using the change of base formula:

$$y = \frac{\log\left(\frac{x}{50,000}\right)}{\log(0.8)}$$

Since $\log(0.8) = \log\left(\frac{1}{10/5}\right) = \log\left(\frac{1}{2}\right)$, we can simplify the equation:

$$y = \frac{\log\left(\frac{x}{50,000}\right)}{\log\left(\frac{1}{2}\right)}$$

Using the property of logarithms that $\log_a(b) = -\log_a(\frac{1}{b})$, we can simplify the equation:

$$y = \frac{\log\left(\frac{x}{50,000}\right)}{-\log\left(\frac{1}{2}\right)}$$

$$y = \frac{\log\left(\frac{x}{50,000}\right)}{\log\left(2\right)}$$

$$y = \log_{2}\left(\frac{x}{50,000}\right)$$

However, we need to express the equation in terms of $\log_{50,000}$, not $\log_{2}$. We can do this by using the change of base formula:

$$y = \frac{\log\left(\frac{x}{50,000}\right)}{\log(50,000)}$$

$$y = \log_{50,000}\left(\frac{x}{50,000}\right)$$

However, we need to express the equation in terms of $\log_{0.8}$, not $\log_{50,000}$. We can do this by using the change of base formula:

$$y = \frac{\log\left(\frac{x}{50,000}\right)}{\log(0.8)}$$

$$y = \log_{0.8}\left(\frac{x}{20,000}\right)$$

Conclusion

After following the steps outlined above, we have found the inverse function of $f(x)=50,000(0.8)^x$ to be $f^{-1}(x)=\log_{0.8}\left(\frac{x}{20,000}\right)$. This is the correct answer among the provided options.

Final Answer

The final answer is: $\boxed{A}$

Introduction

Inverse functions are a fundamental concept in mathematics, particularly in algebra and calculus. They play a crucial role in solving equations, modeling real-world phenomena, and understanding the behavior of functions. In this article, we will delve into the concept of inverse functions, explore the given function $f(x)=50,000(0.8)^x$, and determine the correct inverse function among the provided options. We will also provide a comprehensive Q&A section to help you better understand the concept of inverse functions.

Understanding Inverse Functions

An inverse function is a function that reverses the operation of another function. In other words, if we have a function $f(x)$, its inverse function $f^{-1}(x)$ will take the output of $f(x)$ and return the original input. The inverse function is denoted by $f^{-1}(x)$ and is read as "f inverse of x".

To find the inverse of a function, we need to follow these steps:

1. Replace $f(x)$ with $y$.
2. Swap the roles of $x$ and $y$.
3. Solve for $y$.

The Given Function

The given function is $f(x)=50,000(0.8)^x$. To find its inverse, we will follow the steps outlined above.

Step 1: Replace $f(x)$ with $y$

$$y = 50,000(0.8)^x$$

Step 2: Swap the roles of $x$ and $y$

$$x = 50,000(0.8)^y$$

Step 3: Solve for $y$

To solve for $y$, we need to isolate it on one side of the equation. We can do this by using logarithms.

$$\log_{0.8}(x) = \log_{0.8}(50,000(0.8)^y)$$

Using the property of logarithms that $\log_a(a^b) = b$, we can simplify the equation:

$$\log_{0.8}(x) = y\log_{0.8}(0.8) + \log_{0.8}(50,000)$$

Since $\log_{0.8}(0.8) = 1$, we can further simplify the equation:

$$\log_{0.8}(x) = y + \log_{0.8}(50,000)$$

Now, we can isolate $y$ by subtracting $\log_{0.8}(50,000)$ from both sides:

$$y = \log_{0.8}(x) - \log_{0.8}(50,000)$$

Using the property of logarithms that $\log_a(b) - \log_a(c) = \log_a(\frac{b}{c})$, we can simplify the equation:

$$y = \log_{0.8}\left(\frac{x}{50,000}\right)$$

However, we need to express the equation in terms of $\log_{50,000}$, not $\log_{0.8}$. We can do this by using the change of base formula:

$$y = \frac{\log\left(\frac{x}{50,000}\right)}{\log(0.8)}$$

Since $\log(0.8) = \log\left(\frac{1}{10/5}\right) = \log\left(\frac{1}{2}\right)$, we can simplify the equation:

$$y = \frac{\log\left(\frac{x}{50,000}\right)}{\log\left(\frac{1}{2}\right)}$$

Using the property of logarithms that $\log_a(b) = -\log_a(\frac{1}{b})$, we can simplify the equation:

$$y = \frac{\log\left(\frac{x}{50,000}\right)}{-\log\left(\frac{1}{2}\right)}$$

$$y = \frac{\log\left(\frac{x}{50,000}\right)}{\log\left(2\right)}$$

$$y = \log_{2}\left(\frac{x}{50,000}\right)$$

However, we need to express the equation in terms of $\log_{50,000}$, not $\log_{2}$. We can do this by using the change of base formula:

$$y = \frac{\log\left(\frac{x}{50,000}\right)}{\log(50,000)}$$

$$y = \log_{50,000}\left(\frac{x}{50,000}\right)$$

However, we need to express the equation in terms of $\log_{0.8}$, not $\log_{50,000}$. We can do this by using the change of base formula:

$$y = \frac{\log\left(\frac{x}{50,000}\right)}{\log(0.8)}$$

$$y = \log_{0.8}\left(\frac{x}{20,000}\right)$$

Conclusion

After following the steps outlined above, we have found the inverse function of $f(x)=50,000(0.8)^x$ to be $f^{-1}(x)=\log_{0.8}\left(\frac{x}{20,000}\right)$. This is the correct answer among the provided options.

Q&A Section

Q: What is an inverse function?

A: An inverse function is a function that reverses the operation of another function. In other words, if we have a function $f(x)$, its inverse function $f^{-1}(x)$ will take the output of $f(x)$ and return the original input.

Q: How do I find the inverse of a function?

A: To find the inverse of a function, you need to follow these steps:

1. Replace $f(x)$ with $y$.
2. Swap the roles of $x$ and $y$.
3. Solve for $y$.

Q: What is the given function?

A: The given function is $f(x)=50,000(0.8)^x$.

Q: What is the inverse function of the given function?

A: The inverse function of the given function is $f^{-1}(x)=\log_{0.8}\left(\frac{x}{20,000}\right)$.

Q: Why do we need to use logarithms to find the inverse of a function?

A: We need to use logarithms to find the inverse of a function because it allows us to isolate the variable $y$ on one side of the equation.

Q: What is the change of base formula?

A: The change of base formula is a formula that allows us to change the base of a logarithm from one base to another. It is given by:

$$\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$$

Q: How do I use the change of base formula?

A: To use the change of base formula, you need to follow these steps:

1. Identify the base of the logarithm that you want to change.
2. Identify the new base that you want to use.
3. Plug the values into the formula and simplify.

Q: What is the final answer?

A: The final answer is $\boxed{A}$.