The One-to-one Function $f$ Is Defined Below.$f(x) = (x + 5)^3$Find $f^{-1}(x$\], Where $f^{-1}$ Is The Inverse Of $f$.$f^{-1}(x) = $

Introduction

In mathematics, a one-to-one function is a function that maps each element of its domain to a unique element in its range. This means that no two elements in the domain can map to the same element in the range. In this article, we will explore the concept of a one-to-one function and its inverse. We will define a one-to-one function $f(x) = (x + 5)^3$ and find its inverse $f^{-1}(x)$.

What is a One-to-One Function?

A one-to-one function is a function that satisfies the following condition:

• For every $x_1$ and $x_2$ in the domain of $f$, if $f(x_1) = f(x_2)$, then $x_1 = x_2$.

In other words, a one-to-one function is a function that maps each element of its domain to a unique element in its range. This means that no two elements in the domain can map to the same element in the range.

The One-to-One Function $f(x) = (x + 5)^3$

The one-to-one function $f(x) = (x + 5)^3$ is a cubic function that maps each element of its domain to a unique element in its range. To find the inverse of this function, we need to find a function $f^{-1}(x)$ such that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Finding the Inverse of $f(x) = (x + 5)^3$

To find the inverse of $f(x) = (x + 5)^3$, we need to solve the equation $y = (x + 5)^3$ for $x$. This will give us the inverse function $f^{-1}(x)$.

Step 1: Take the cube root of both sides

The first step in finding the inverse of $f(x) = (x + 5)^3$ is to take the cube root of both sides of the equation:

$$y = (x + 5)^3$$

$$\sqrt[3]{y} = \sqrt[3]{(x + 5)^3}$$

$$\sqrt[3]{y} = x + 5$$

Step 2: Subtract 5 from both sides

The next step is to subtract 5 from both sides of the equation:

$$\sqrt[3]{y} - 5 = x$$

Step 3: Replace $y$ with $x$

The final step is to replace $y$ with $x$ in the equation:

$$\sqrt[3]{x} - 5 = f^{-1}(x)$$

The Inverse Function $f^{-1}(x)$

The inverse function $f^{-1}(x)$ is given by:

$$f^{-1}(x) = \sqrt[3]{x} - 5$$

Conclusion

In this article, we defined a one-to-one function $f(x) = (x + 5)^3$ and found its inverse $f^{-1}(x)$. We used the concept of a one-to-one function and the definition of an inverse function to find the inverse of $f(x) = (x + 5)^3$. The inverse function $f^{-1}(x)$ is given by:

$$f^{-1}(x) = \sqrt[3]{x} - 5$$

Example Problems

Here are some example problems to help you practice finding the inverse of a one-to-one function:

Example 1

Find the inverse of the function $f(x) = (x - 2)^2$.

Solution

To find the inverse of $f(x) = (x - 2)^2$, we need to solve the equation $y = (x - 2)^2$ for $x$. This will give us the inverse function $f^{-1}(x)$.

Step 1: Take the square root of both sides

The first step in finding the inverse of $f(x) = (x - 2)^2$ is to take the square root of both sides of the equation:

$$y = (x - 2)^2$$

$$\sqrt{y} = \sqrt{(x - 2)^2}$$

$$\sqrt{y} = x - 2$$

Step 2: Add 2 to both sides

The next step is to add 2 to both sides of the equation:

$$\sqrt{y} + 2 = x$$

Step 3: Replace $y$ with $x$

The final step is to replace $y$ with $x$ in the equation:

$$\sqrt{x} + 2 = f^{-1}(x)$$

Example 2

Find the inverse of the function $f(x) = (x + 1)^4$.

Solution

To find the inverse of $f(x) = (x + 1)^4$, we need to solve the equation $y = (x + 1)^4$ for $x$. This will give us the inverse function $f^{-1}(x)$.

Step 1: Take the fourth root of both sides

The first step in finding the inverse of $f(x) = (x + 1)^4$ is to take the fourth root of both sides of the equation:

$$y = (x + 1)^4$$

$$\sqrt[4]{y} = \sqrt[4]{(x + 1)^4}$$

$$\sqrt[4]{y} = x + 1$$

Step 2: Subtract 1 from both sides

The next step is to subtract 1 from both sides of the equation:

$$\sqrt[4]{y} - 1 = x$$

Step 3: Replace $y$ with $x$

The final step is to replace $y$ with $x$ in the equation:

$$\sqrt[4]{x} - 1 = f^{-1}(x)$$

Conclusion

In this article, we defined a one-to-one function $f(x) = (x + 5)^3$ and found its inverse $f^{-1}(x)$. We used the concept of a one-to-one function and the definition of an inverse function to find the inverse of $f(x) = (x + 5)^3$. The inverse function $f^{-1}(x)$ is given by:

$$f^{-1}(x) = \sqrt[3]{x} - 5$$

Q: What is a one-to-one function?

A: A one-to-one function is a function that maps each element of its domain to a unique element in its range. This means that no two elements in the domain can map to the same element in the range.

Q: How do I know if a function is one-to-one?

A: To determine if a function is one-to-one, you can use the following criteria:

• For every $x_1$ and $x_2$ in the domain of $f$, if $f(x_1) = f(x_2)$, then $x_1 = x_2$.

In other words, a function is one-to-one if it maps each element of its domain to a unique element in its range.

Q: What is the inverse of a one-to-one function?

A: The inverse of a one-to-one function $f(x)$ is a function $f^{-1}(x)$ such that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Q: How do I find the inverse of a one-to-one function?

A: To find the inverse of a one-to-one function, you can follow these steps:

1. Write the equation $y = f(x)$.
2. Swap the variables $x$ and $y$ to get $x = f(y)$.
3. Solve the equation $x = f(y)$ for $y$.
4. Replace $y$ with $f^{-1}(x)$ to get the inverse function.

Q: What is the difference between a one-to-one function and a function with an inverse?

A: A one-to-one function is a function that maps each element of its domain to a unique element in its range. A function with an inverse is a function that has an inverse function, which is a function that maps each element of its range to a unique element in its domain.

Q: Can a function have an inverse if it is not one-to-one?

A: No, a function cannot have an inverse if it is not one-to-one. The inverse of a function is a function that maps each element of its range to a unique element in its domain. If a function is not one-to-one, it does not have a unique inverse.

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

A: To determine if a function has an inverse, you can use the following criteria:

• The function must be one-to-one.
• The function must be defined for all values of $x$ in its domain.
• The function must be continuous for all values of $x$ in its domain.

Q: What is the significance of one-to-one functions and their inverses in mathematics?

A: One-to-one functions and their inverses are important in mathematics because they allow us to solve equations and inequalities. They also help us to understand the properties of functions and how they behave.

Q: Can one-to-one functions and their inverses be used in real-world applications?

A: Yes, one-to-one functions and their inverses can be used in real-world applications. For example, they can be used to model population growth, chemical reactions, and electrical circuits.

Q: How do I apply one-to-one functions and their inverses in real-world applications?

A: To apply one-to-one functions and their inverses in real-world applications, you can follow these steps:

1. Identify the problem you want to solve.
2. Determine the one-to-one function that models the problem.
3. Find the inverse of the one-to-one function.
4. Use the inverse function to solve the problem.

Conclusion

In this article, we answered some common questions about one-to-one functions and their inverses. We discussed the definition of a one-to-one function, how to determine if a function is one-to-one, and how to find the inverse of a one-to-one function. We also discussed the significance of one-to-one functions and their inverses in mathematics and how they can be used in real-world applications.