QUESTIONS51. The Graph Of $f(x)=u^{x-1}$ Passes Through The Point \left(0, \frac{1}{2}\right ]. Calculate The Value Of U U U .52. Write Down The Equation Of The Inverse Of F F F In The Form
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.
Related Topics
- Exponential functions
- Inverse functions
- Logarithmic functions
- Mathematical modeling
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.