Let F ( X ) = X 2 − 2 X + 1 F(x) = X^2 - 2x + 1 F ( X ) = X 2 − 2 X + 1 . Find The Inverse Function Of F F F By Identifying An Appropriate Restriction Of Its Domain.A. The Inverse Is F − 1 ( X ) = X + 1 F^{-1}(x) = \sqrt{x+1} F − 1 ( X ) = X + 1 ​ If The Domain Of F F F Is Restricted To $x \geq

Introduction

In mathematics, the concept of inverse functions is crucial in understanding the relationship between two functions. Given a function $f(x)$, its inverse function $f^{-1}(x)$ is a function that undoes the action of $f(x)$. In other words, if $f(x)$ maps an input $x$ to an output $y$, then $f^{-1}(x)$ maps the output $y$ back to the input $x$. In this article, we will explore how to find the inverse function of a quadratic function, specifically the function $f(x) = x^2 - 2x + 1$.

Understanding the Function

The given function is a quadratic function in the form of $f(x) = ax^2 + bx + c$, where $a = 1$, $b = -2$, and $c = 1$. To find the inverse function, we need to first understand the behavior of this function. The graph of the function is a parabola that opens upwards, with its vertex at the point $(1, 0)$. This means that the function is increasing on the left side of the vertex and decreasing on the right side.

Finding the Inverse Function

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

1. Switch the roles of x and y: We start by switching the roles of $x$ and $y$ in the original function. This gives us the equation $x = y^2 - 2y + 1$.
2. Solve for y: We need to solve this equation for $y$. To do this, we can use the quadratic formula: $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
3. Simplify the expression: After solving for $y$, we need to simplify the expression to get the inverse function.

Solving for y

Using the quadratic formula, we get:

$$y = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(1)}}{2(1)}$$

$$y = \frac{2 \pm \sqrt{4 - 4}}{2}$$

$$y = \frac{2 \pm \sqrt{0}}{2}$$

$$y = \frac{2}{2}$$

$$y = 1$$

Restricting the Domain

However, we notice that the equation $y = 1$ is not the inverse function of $f(x) = x^2 - 2x + 1$. This is because the original function is a quadratic function, and its inverse function is not a simple linear function. To find the inverse function, we need to restrict the domain of the original function.

Restricting the Domain to x ≥ 1

If we restrict the domain of the original function to $x \geq 1$, then the inverse function is given by:

$$f^{-1}(x) = \sqrt{x+1}$$

This is because the original function is increasing on the left side of the vertex, and the inverse function is decreasing on the right side of the vertex.

Conclusion

In conclusion, finding the inverse function of a quadratic function requires us to restrict the domain of the original function. By restricting the domain to $x \geq 1$, we can find the inverse function of $f(x) = x^2 - 2x + 1$, which is given by $f^{-1}(x) = \sqrt{x+1}$. This inverse function is a square root function that maps the output $x$ back to the input $y$.

Example

Let's consider an example to illustrate this concept. Suppose we have a function $f(x) = x^2 - 2x + 1$ and we want to find its inverse function. We can use the steps outlined above to find the inverse function.

Step 1: Switch the roles of x and y

We start by switching the roles of $x$ and $y$ in the original function. This gives us the equation $x = y^2 - 2y + 1$.

Step 2: Solve for y

We need to solve this equation for $y$. To do this, we can use the quadratic formula: $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Step 3: Simplify the expression

After solving for $y$, we need to simplify the expression to get the inverse function.

Step 4: Restrict the domain

We need to restrict the domain of the original function to $x \geq 1$.

Step 5: Find the inverse function

Using the steps above, we can find the inverse function of $f(x) = x^2 - 2x + 1$, which is given by $f^{-1}(x) = \sqrt{x+1}$.

Applications

The concept of inverse functions has many applications in mathematics and other fields. For example, in physics, the inverse function is used to describe the relationship between the position and velocity of an object. In engineering, the inverse function is used to design control systems that can adjust the output of a system based on the input.

Conclusion

$$$$$$$$$$

Q: What is an inverse function?

A: An inverse function is a function that undoes the action of another function. In other words, if a function $f(x)$ maps an input $x$ to an output $y$, then its inverse function $f^{-1}(x)$ maps the output $y$ back to the input $x$.

Q: Why do we need inverse functions?

A: Inverse functions are essential in many areas of mathematics and other fields. They help us to solve equations, model real-world phenomena, and make predictions. Inverse functions also help us to understand the relationship between two functions and to identify the input and output values of a function.

Q: How do we find the inverse function of a quadratic function?

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

1. Switch the roles of x and y: We start by switching the roles of $x$ and $y$ in the original function. This gives us the equation $x = y^2 - 2y + 1$.
2. Solve for y: We need to solve this equation for $y$. To do this, we can use the quadratic formula: $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
3. Simplify the expression: After solving for $y$, we need to simplify the expression to get the inverse function.
4. Restrict the domain: We need to restrict the domain of the original function to $x \geq 1$.

Q: What is the inverse function of $f(x) = x^2 - 2x + 1$?

A: The inverse function of $f(x) = x^2 - 2x + 1$ is given by $f^{-1}(x) = \sqrt{x+1}$.

Q: Why do we need to restrict the domain of the original function?

A: We need to restrict the domain of the original function to ensure that the inverse function is well-defined. In this case, we restricted the domain to $x \geq 1$ to ensure that the inverse function is a square root function.

Q: What are some applications of inverse functions?

A: Inverse functions have many applications in mathematics and other fields. Some examples include:

• Physics: Inverse functions are used to describe the relationship between the position and velocity of an object.
• Engineering: Inverse functions are used to design control systems that can adjust the output of a system based on the input.
• Computer Science: Inverse functions are used in algorithms and data structures to solve problems and make predictions.

Q: How do we use inverse functions in real-world problems?

A: Inverse functions are used in many real-world problems, such as:

• Optimization: Inverse functions are used to optimize functions and find the maximum or minimum value of a function.
• Modeling: Inverse functions are used to model real-world phenomena, such as population growth and disease spread.
• Prediction: Inverse functions are used to make predictions about future events, such as weather forecasting and stock market analysis.

Q: What are some common mistakes to avoid when working with inverse functions?

A: Some common mistakes to avoid when working with inverse functions include:

• Not restricting the domain: Failing to restrict the domain of the original function can lead to undefined or incorrect inverse functions.
• Not simplifying the expression: Failing to simplify the expression for the inverse function can lead to incorrect or complex inverse functions.
• Not checking the domain: Failing to check the domain of the inverse function can lead to undefined or incorrect inverse functions.

Q: How do we check the domain of an inverse function?

A: To check the domain of an inverse function, we need to ensure that the inverse function is well-defined and that the input values are valid. We can do this by:

• Checking the range: Checking the range of the original function to ensure that it is valid for the inverse function.
• Checking the domain: Checking the domain of the original function to ensure that it is valid for the inverse function.
• Simplifying the expression: Simplifying the expression for the inverse function to ensure that it is well-defined and valid.