Comparing A Function And Its InverseThe Average Cost Per Person Of Attending An Online Class Depends On The Number Of People Attending. - If 20 People Attend, The Web Host Charges $$ 10$ Per Person. If 26 People Attend, The Web Host
Understanding the Concept of Inverse Functions
In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). An inverse function is a function that reverses the operation of the original function. In other words, if a function takes an input and produces an output, the inverse function takes the output and produces the original input.
The Average Cost Function
Let's consider a real-world example to illustrate the concept of inverse functions. Suppose we have an online class that charges a web host based on the number of people attending. The average cost per person depends on the number of people attending. If 20 people attend, the web host charges $10 per person. If 26 people attend, the web host charges $8 per person. We can represent this situation using a function, which we'll call f(x), where x is the number of people attending and f(x) is the average cost per person.
Defining the Function
Let's define the function f(x) as follows:
f(x) = 10 - (x - 20) / 5
This function takes the number of people attending (x) as input and produces the average cost per person (f(x)) as output.
Finding the Inverse Function
To find the inverse function, we need to swap the input and output variables. In other words, we need to find a new function, which we'll call f^(-1)(x), that takes the average cost per person (x) as input and produces the number of people attending (f^(-1)(x)) as output.
To find the inverse function, we can use the following steps:
- Swap the input and output variables: x = f(x) becomes f(x) = x.
- Solve for x: f(x) = 10 - (x - 20) / 5 becomes x = 10 - (f(x) - 20) / 5.
- Simplify the equation: x = 10 - (f(x) - 20) / 5 becomes x = 10 - (f(x) - 20) / 5.
Solving for the Inverse Function
To solve for the inverse function, we can use algebraic manipulation. Let's start by simplifying the equation:
x = 10 - (f(x) - 20) / 5
We can multiply both sides of the equation by 5 to eliminate the fraction:
5x = 50 - (f(x) - 20)
Next, we can add f(x) to both sides of the equation:
5x + f(x) = 70
Now, we can factor out f(x):
f(x)(5 + 1) = 70
Simplifying the equation, we get:
f(x)(6) = 70
Dividing both sides of the equation by 6, we get:
f(x) = 70 / 6
Simplifying the equation, we get:
f(x) = 35 / 3
The Inverse Function
Now that we have found the inverse function, we can represent it as follows:
f^(-1)(x) = 35 / 3
This function takes the average cost per person (x) as input and produces the number of people attending (f^(-1)(x)) as output.
Comparing the Original and Inverse Functions
Let's compare the original function f(x) = 10 - (x - 20) / 5 with the inverse function f^(-1)(x) = 35 / 3.
We can see that the original function takes the number of people attending (x) as input and produces the average cost per person (f(x)) as output. The inverse function takes the average cost per person (x) as input and produces the number of people attending (f^(-1)(x)) as output.
Graphing the Functions
To visualize the functions, we can graph them on a coordinate plane.
The original function f(x) = 10 - (x - 20) / 5 is a linear function that passes through the points (20, 10) and (26, 8).
The inverse function f^(-1)(x) = 35 / 3 is also a linear function that passes through the points (10, 20) and (8, 26).
Conclusion
In this article, we have compared a function and its inverse. We have defined the original function f(x) = 10 - (x - 20) / 5 and found its inverse function f^(-1)(x) = 35 / 3. We have also graphed the functions on a coordinate plane to visualize their behavior.
Key Takeaways
- A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
- An inverse function is a function that reverses the operation of the original function.
- To find the inverse function, we need to swap the input and output variables and solve for the new variable.
- The inverse function takes the output of the original function as input and produces the original input as output.
References
- [1] "Functions and Inverse Functions" by Khan Academy
- [2] "Inverse Functions" by Math Is Fun
- [3] "Functions and Graphs" by Wolfram MathWorld
Q&A: Comparing a Function and Its Inverse =============================================
Frequently Asked Questions
In this article, we will answer some frequently asked questions about comparing a function and its inverse.
Q: What is a function?
A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). It takes an input and produces an output.
Q: What is an inverse function?
A: An inverse function is a function that reverses the operation of the original function. It takes the output of the original function as input and produces the original input as output.
Q: How do I find the inverse function?
A: To find the inverse function, you need to swap the input and output variables and solve for the new variable. You can use algebraic manipulation to simplify the equation.
Q: What is the difference between a function and its inverse?
A: The main difference between a function and its inverse is that the function takes an input and produces an output, while the inverse function takes the output of the function as input and produces the original input as output.
Q: Can a function have multiple inverses?
A: No, a function can only have one inverse. If a function has multiple inverses, it is not a one-to-one function.
Q: How do I graph a function and its inverse?
A: To graph a function and its inverse, you can use a coordinate plane. The function will be represented by a curve, and the inverse function will be represented by a curve that is symmetric to the original curve.
Q: What are some real-world examples of functions and their inverses?
A: Some real-world examples of functions and their inverses include:
- The cost of a product versus the number of products sold
- The time it takes to complete a task versus the number of tasks completed
- The distance traveled versus the speed of travel
Q: How do I use functions and their inverses in real-world applications?
A: Functions and their inverses can be used in a variety of real-world applications, including:
- Modeling population growth
- Predicting stock prices
- Optimizing resource allocation
Q: What are some common mistakes to avoid when working with functions and their inverses?
A: Some common mistakes to avoid when working with functions and their inverses include:
- Not checking for one-to-one functions
- Not swapping the input and output variables when finding the inverse function
- Not simplifying the equation when finding the inverse function
Conclusion
In this article, we have answered some frequently asked questions about comparing a function and its inverse. We have discussed the definition of a function and its inverse, how to find the inverse function, and how to graph a function and its inverse. We have also provided some real-world examples of functions and their inverses and discussed how to use them in real-world applications.
Key Takeaways
- A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
- An inverse function is a function that reverses the operation of the original function.
- To find the inverse function, you need to swap the input and output variables and solve for the new variable.
- Functions and their inverses can be used in a variety of real-world applications.
References
- [1] "Functions and Inverse Functions" by Khan Academy
- [2] "Inverse Functions" by Math Is Fun
- [3] "Functions and Graphs" by Wolfram MathWorld