Writing Down the Equation of the Inverse of f

Oct 13, 2025 by windailys.com 94 views

Introduction

In this article, we will explore the concept of exponential functions and how to calculate the value of a variable in a given function. We will focus on the graph of the function f(x) = u^(x-1) and use the point (0, 1/2) to determine the value of u.

Understanding the Function f(x) = u^(x-1)

The function f(x) = u^(x-1) is an exponential function where the base is u and the exponent is x-1. This function can be rewritten as f(x) = u^x / u, which is a combination of two exponential functions.

Using the Point (0, 1/2) to Calculate the Value of u

We are given that the graph of f(x) = u^(x-1) passes through the point (0, 1/2). This means that when x = 0, the value of f(x) is 1/2. We can substitute these values into the function to get:

f(0) = u^(0-1) f(0) = u^(-1) f(0) = 1/u f(0) = 1/2

Solving for u

Now we have the equation 1/u = 1/2. To solve for u, we can multiply both sides of the equation by u to get:

1 = u/2 2 = u

Conclusion

Therefore, the value of u in the graph of f(x) = u^(x-1) is 2.

Writing Down the Equation of the Inverse of f

Introduction

In this section, we will explore the concept of inverse functions and how to write down the equation of the inverse of a given function. We will focus on the function f(x) = u^(x-1) and use the value of u to determine the equation of its inverse.

Understanding the Inverse of a Function

The inverse of a function is a function that undoes the action of the original function. In other words, if we have a function f(x) and its inverse f^(-1)(x), then f(f^(-1)(x)) = x and f^(-1)(f(x)) = x.

Finding the Inverse of f(x) = u^(x-1)

To find the inverse of f(x) = u^(x-1), we can start by writing y = u^(x-1). Then, we can switch the roles of x and y to get x = u^(y-1).

Solving for y

Now we have the equation x = u^(y-1). To solve for y, we can take the logarithm of both sides of the equation to get:

log(x) = log(u^(y-1)) log(x) = (y-1)log(u) log(x) = ylog(u) - log(u) log(x) + log(u) = ylog(u) log(x) + log(u) / log(u) = y log(x) + 1 = y

Conclusion

Therefore, the equation of the inverse of f(x) = u^(x-1) is y = log(x) + 1.

Final Answer

The value of u in the graph of f(x) = u^(x-1) is 2, and the equation of the inverse of f is y = log(x) + 1.

Discussion

The concept of exponential functions and inverse functions is a fundamental part of mathematics. In this article, we have explored how to calculate the value of a variable in a given function and how to write down the equation of the inverse of a function. We have used the point (0, 1/2) to determine the value of u in the graph of f(x) = u^(x-1) and have found that the equation of the inverse of f is y = log(x) + 1.

References

[1] "Exponential Functions" by Math Open Reference
[2] "Inverse Functions" by Math Open Reference
[3] "Logarithmic Functions" by Math Open Reference

Introduction

In our previous article, we explored the concept of exponential functions and how to calculate the value of a variable in a given function. We focused on the graph of the function f(x) = u^(x-1) and used the point (0, 1/2) to determine the value of u. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the value of u in the graph of f(x) = u^(x-1)?

A: The value of u in the graph of f(x) = u^(x-1) is 2.

Q: How did you calculate the value of u?

A: We used the point (0, 1/2) to calculate the value of u. We substituted x = 0 and f(x) = 1/2 into the function f(x) = u^(x-1) and solved for u.

Q: What is the equation of the inverse of f(x) = u^(x-1)?

A: The equation of the inverse of f(x) = u^(x-1) is y = log(x) + 1.

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

A: To find the inverse of a function, you can start by writing y = f(x). Then, you can switch the roles of x and y to get x = f(y). Finally, you can solve for y to get the equation of the inverse.

Q: What is the relationship between exponential functions and logarithmic functions?

A: Exponential functions and logarithmic functions are inverse functions of each other. This means that if you have an exponential function f(x) = a^x, then its inverse is f^(-1)(x) = log_a(x).

Q: Can you give an example of how to use the equation of the inverse of f(x) = u^(x-1)?

A: Yes, let's say we want to find the value of x when y = 1. We can substitute y = 1 into the equation of the inverse of f(x) = u^(x-1), which is y = log(x) + 1. This gives us:

1 = log(x) + 1 0 = log(x) x = e^0 x = 1

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

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

Modeling population growth and decay
Calculating interest rates and investments
Analyzing data in fields such as medicine, economics, and social sciences
Solving problems in physics and engineering

Q: Can you recommend any resources for learning more about exponential functions and inverse functions?

A: Yes, here are some resources that you may find helpful:

Online tutorials and videos, such as Khan Academy and 3Blue1Brown
Textbooks and workbooks, such as "Calculus" by Michael Spivak and "Algebra" by Michael Artin
Online courses and degree programs, such as Coursera and edX

Conclusion

In this article, we have answered some frequently asked questions related to calculating the value of u in the graph of f(x) = u^(x-1). We have also discussed the equation of the inverse of f(x) = u^(x-1) and provided some real-world applications of exponential functions and inverse functions. We hope that this article has been helpful in clarifying any questions you may have had about this topic.