Given $f(x)=\left(\frac{2}{3}\right) X$, Which One Of The Following Is The Graph Of $f^{-1}(x)$?

by ADMIN 101 views

Inverse functions are a crucial concept in mathematics, particularly in algebra and calculus. They play a vital role in solving equations, modeling real-world phenomena, and understanding the behavior of functions. In this article, we will explore the concept of inverse functions and graphs, focusing on the given function $f(x)=\left(\frac{2}{3}\right) x$ and determining which one of the following is the graph of $f^{-1}(x)$.

What are 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 $f^{-1}(x)$ undoes the operation of $f(x)$. This means that if we apply $f(x)$ to a value and then apply $f^{-1}(x)$ to the result, we will get back the original value.

Properties of Inverse Functions

Inverse functions have several important properties that make them useful in mathematics. Some of these properties include:

  • One-to-One Correspondence: An inverse function is a one-to-one correspondence between the domain and range of the original function.
  • Symmetry: The graph of an inverse function is symmetric to the graph of the original function with respect to the line $y=x$.
  • Reversibility: If we apply an inverse function to a value, we will get back the original value.

Finding the Inverse of a Function

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

  1. Replace $f(x)$ with $y$: We start by replacing the function $f(x)$ with $y$.
  2. Interchange $x$ and $y$: We interchange the variables $x$ and $y$ to get $x=f^{-1}(y)$.
  3. Solve for $y$: We solve for $y$ to get the inverse function $f^{-1}(x)$.

Finding the Inverse of $f(x)=\left(\frac{2}{3}\right) x$

Now, let's find the inverse of the given function $f(x)=\left(\frac{2}{3}\right) x$. We will follow the steps outlined above.

  1. Replace $f(x)$ with $y$: We start by replacing the function $f(x)$ with $y$ to get $y=\left(\frac{2}{3}\right) x$.
  2. Interchange $x$ and $y$: We interchange the variables $x$ and $y$ to get $x=\left(\frac{2}{3}\right) y$.
  3. Solve for $y$: We solve for $y$ to get $y=\frac{3}{2} x$.

Therefore, the inverse function of $f(x)=\left(\frac{2}{3}\right) x$ is $f^{-1}(x)=\frac{3}{2} x$.

Graph of $f^{-1}(x)$

To determine which one of the following is the graph of $f^{-1}(x)$, we need to analyze the graph of $f(x)=\left(\frac{2}{3}\right) x$ and its inverse function $f^{-1}(x)=\frac{3}{2} x$.

The graph of $f(x)=\left(\frac{2}{3}\right) x$ is a straight line with a slope of $\frac{2}{3}$ and a y-intercept of $0$. The graph of $f^{-1}(x)=\frac{3}{2} x$ is also a straight line with a slope of $\frac{3}{2}$ and a y-intercept of $0$.

Since the graph of $f^{-1}(x)$ is symmetric to the graph of $f(x)$ with respect to the line $y=x$, we can conclude that the graph of $f^{-1}(x)$ is the line $y=\frac{3}{2} x$.

Conclusion

In conclusion, the inverse function of $f(x)=\left(\frac{2}{3}\right) x$ is $f^{-1}(x)=\frac{3}{2} x$, and the graph of $f^{-1}(x)$ is the line $y=\frac{3}{2} x$. This article has provided a comprehensive understanding of inverse functions and graphs, focusing on the given function $f(x)=\left(\frac{2}{3}\right) x$ and determining which one of the following is the graph of $f^{-1}(x)$.

References

  • [1] Calculus by Michael Spivak
  • [2] Algebra by Michael Artin
  • [3] Graph Theory by Douglas B. West

Further Reading

For further reading on inverse functions and graphs, we recommend the following resources:

  • Inverse Functions by Khan Academy
  • Graphs of Inverse Functions by Mathway
  • Inverse Functions and Graphs by Wolfram MathWorld
    Inverse Functions and Graphs: Q&A =====================================

In this article, we will continue to explore the concept of inverse functions and graphs, focusing on the given function $f(x)=\left(\frac{2}{3}\right) x$ and its inverse function $f^{-1}(x)=\frac{3}{2} x$. We will answer some frequently asked questions about inverse functions and graphs, providing a comprehensive understanding of this important mathematical concept.

Q: What is the difference between a function and its inverse?

A: A function and its inverse are two related functions that undo each other's operations. In other words, if we apply a function to a value and then apply its inverse to the result, we will get back the original value.

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 $y$: We start by replacing the function $f(x)$ with $y$.
  2. Interchange $x$ and $y$: We interchange the variables $x$ and $y$ to get $x=f^{-1}(y)$.
  3. Solve for $y$: We solve for $y$ to get the inverse function $f^{-1}(x)$.

Q: What is the relationship between the graph of a function and its inverse?

A: The graph of a function and its inverse are symmetric with respect to the line $y=x$. This means that if we reflect the graph of a function across the line $y=x$, we will get the graph of its inverse.

Q: Can you give an example of a function and its inverse?

A: Yes, let's consider the function $f(x)=\left(\frac{2}{3}\right) x$. Its inverse function is $f^{-1}(x)=\frac{3}{2} x$. We can verify that these two functions are inverses of each other by applying them to a value and checking that we get back the original value.

Q: How do I determine if a function has an inverse?

A: A function has an inverse if it is one-to-one, meaning that each value in the domain maps to a unique value in the range. In other words, a function has an inverse if it passes the horizontal line test.

Q: Can you explain the concept of symmetry in the context of inverse functions?

A: Yes, the concept of symmetry is crucial in understanding inverse functions. The graph of a function and its inverse are symmetric with respect to the line $y=x$. This means that if we reflect the graph of a function across the line $y=x$, we will get the graph of its inverse.

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

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

  1. Graph the original function: We start by graphing the original function.
  2. Reflect the graph across the line $y=x$: We reflect the graph of the original function across the line $y=x$ to get the graph of its inverse.

Q: Can you give an example of graphing the inverse of a function?

A: Yes, let's consider the function $f(x)=\left(\frac{2}{3}\right) x$. We can graph this function and then reflect its graph across the line $y=x$ to get the graph of its inverse.

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

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

  • Modeling real-world phenomena: Inverse functions can be used to model real-world phenomena, such as population growth, chemical reactions, and electrical circuits.
  • Solving equations: Inverse functions can be used to solve equations, such as quadratic equations and systems of equations.
  • Graphing functions: Inverse functions can be used to graph functions, such as linear functions and quadratic functions.

Conclusion

In conclusion, inverse functions and graphs are a fundamental concept in mathematics, with many real-world applications. We have answered some frequently asked questions about inverse functions and graphs, providing a comprehensive understanding of this important mathematical concept. We hope that this article has been helpful in understanding the concept of inverse functions and graphs.

References

  • [1] Calculus by Michael Spivak
  • [2] Algebra by Michael Artin
  • [3] Graph Theory by Douglas B. West

Further Reading

For further reading on inverse functions and graphs, we recommend the following resources:

  • Inverse Functions by Khan Academy
  • Graphs of Inverse Functions by Mathway
  • Inverse Functions and Graphs by Wolfram MathWorld