The Function $f(x) = (x+4)^2$ Is Not One-to-one. Choose The Largest Possible Domain Containing The Number 100 So That The Function Restricted To The Domain Is One-to-one.The Largest Possible Domain Is $(-4, \infty$\]; The Inverse

Introduction

In mathematics, a one-to-one function is a function that maps distinct elements of its domain to distinct elements of its range. In other words, it is a function that never takes on the same value twice. The function $f(x) = (x+4)^2$ is a quadratic function that is not one-to-one because it has a repeated value at $x = -4$. In this article, we will explore the largest possible domain containing the number 100 so that the function restricted to the domain is one-to-one.

Understanding One-to-One Functions

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

$$f(x_1) = f(x_2) \implies x_1 = x_2$$

This means that if the function takes on the same value at two different points, then those two points must be the same. In other words, a one-to-one function never takes on the same value twice.

The Function $f(x) = (x+4)^2$

The function $f(x) = (x+4)^2$ is a quadratic function that is not one-to-one because it has a repeated value at $x = -4$. To see why, let's evaluate the function at $x = -4$ and $x = 0$.

$$f(-4) = (-4+4)^2 = 0^2 = 0$$

$$f(0) = (0+4)^2 = 4^2 = 16$$

As we can see, the function takes on the same value at $x = -4$ and $x = 0$, which means that it is not one-to-one.

Finding the Largest Possible Domain

To find the largest possible domain containing the number 100 so that the function restricted to the domain is one-to-one, we need to find the values of $x$ for which the function is strictly increasing or strictly decreasing.

Let's consider the function $f(x) = (x+4)^2$. We can rewrite it as $f(x) = x^2 + 8x + 16$. To find the values of $x$ for which the function is strictly increasing or strictly decreasing, we need to find the values of $x$ for which the derivative of the function is positive or negative.

The derivative of the function is given by:

$$f'(x) = 2x + 8$$

To find the values of $x$ for which the derivative is positive or negative, we can set the derivative equal to zero and solve for $x$.

$$2x + 8 = 0 \implies x = -4$$

As we can see, the derivative is positive for $x > -4$ and negative for $x < -4$. This means that the function is strictly increasing for $x > -4$ and strictly decreasing for $x < -4$.

The Largest Possible Domain

Based on the analysis above, the largest possible domain containing the number 100 so that the function restricted to the domain is one-to-one is $(-4, \infty)$.

The Inverse Function

To find the inverse function, we need to swap the roles of $x$ and $y$ and solve for $y$.

$$y = (x+4)^2 \implies x = (y+4)^2$$

Solving for $y$, we get:

$$y = \pm \sqrt{x+4} - 4$$

Since the function is strictly increasing for $x > -4$, we can take the positive square root.

$$y = \sqrt{x+4} - 4$$

This is the inverse function.

Conclusion

Introduction

In our previous article, we explored the function $f(x) = (x+4)^2$ and its inverse. We found that the largest possible domain containing the number 100 so that the function restricted to the domain is one-to-one is $(-4, \infty)$. In this article, we will answer some frequently asked questions about the function and its inverse.

Q: What is the domain of the function $f(x) = (x+4)^2$?

A: The domain of the function $f(x) = (x+4)^2$ is all real numbers, i.e., $(-\infty, \infty)$.

Q: Why is the function $f(x) = (x+4)^2$ not one-to-one?

A: The function $f(x) = (x+4)^2$ is not one-to-one because it has a repeated value at $x = -4$. This means that the function takes on the same value at two different points, which is not allowed in a one-to-one function.

Q: How do we find the inverse function of $f(x) = (x+4)^2$?

A: To find the inverse function of $f(x) = (x+4)^2$, we need to swap the roles of $x$ and $y$ and solve for $y$. This gives us the equation $x = (y+4)^2$. Solving for $y$, we get $y = \pm \sqrt{x+4} - 4$. Since the function is strictly increasing for $x > -4$, we can take the positive square root.

Q: What is the range of the inverse function $y = \sqrt{x+4} - 4$?

A: The range of the inverse function $y = \sqrt{x+4} - 4$ is all real numbers greater than or equal to $-4$, i.e., $[-4, \infty)$.

Q: How do we know that the inverse function $y = \sqrt{x+4} - 4$ is one-to-one?

A: We know that the inverse function $y = \sqrt{x+4} - 4$ is one-to-one because it is the inverse of the function $f(x) = (x+4)^2$, which is not one-to-one. This means that the inverse function is also not one-to-one, and therefore it is one-to-one.

Q: Can we find the inverse function of $f(x) = (x+4)^2$ using other methods?

A: Yes, we can find the inverse function of $f(x) = (x+4)^2$ using other methods, such as using the quadratic formula or using a graphing calculator. However, the method we used in this article is the most straightforward and easiest to understand.

Q: What are some real-world applications of the function $f(x) = (x+4)^2$ and its inverse?

A: The function $f(x) = (x+4)^2$ and its inverse have many real-world applications, such as modeling population growth, modeling the spread of diseases, and modeling the behavior of physical systems. The inverse function can be used to find the original input that produced a given output, which is useful in many applications.

Conclusion

In conclusion, the function $f(x) = (x+4)^2$ and its inverse have many interesting properties and applications. We hope that this article has helped to clarify some of the concepts and has provided a better understanding of the function and its inverse.