For The Function $f(x) = 3x^2 - 2, \, X \geq 0$, Answer The Following Questions:a) Determine Whether $f$ Is One-to-one. B) If The Function Is One-to-one, Find A Formula For The Inverse.Select The Correct Choice Below, And If

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. Inverse functions, on the other hand, are functions that reverse the operation of the original function. In this article, we will analyze the function $f(x) = 3x^2 - 2, \, x \geq 0$ and determine whether it is one-to-one. If it is one-to-one, we will find a formula for the inverse.

One-to-One Functions

A function $f$ is one-to-one if and only if it satisfies the following condition:

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

This means that if $f(x_1) = f(x_2)$, then $x_1$ must equal $x_2$. In other words, no two elements in the domain can map to the same element in the range.

The Function $f(x) = 3x^2 - 2, \, x \geq 0$

The function $f(x) = 3x^2 - 2, \, x \geq 0$ is a quadratic function that is defined for all non-negative real numbers. To determine whether this function is one-to-one, we need to examine its graph.

Graph of the Function

The graph of the function $f(x) = 3x^2 - 2, \, x \geq 0$ is a parabola that opens upwards. Since the function is defined for all non-negative real numbers, the graph will only be visible in the first quadrant of the coordinate plane.

Is the Function One-to-One?

To determine whether the function is one-to-one, we need to examine the graph and see if it passes the horizontal line test. If the graph passes the horizontal line test, then the function is one-to-one.

Horizontal Line Test

The horizontal line test states that if a function is one-to-one, then no horizontal line will intersect the graph of the function more than once. In other words, if a horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one.

Applying the Horizontal Line Test

Let's apply the horizontal line test to the graph of the function $f(x) = 3x^2 - 2, \, x \geq 0$. We can draw a horizontal line at any point on the graph and see if it intersects the graph at more than one point.

Conclusion

After applying the horizontal line test, we can conclude that the function $f(x) = 3x^2 - 2, \, x \geq 0$ is not one-to-one. This is because the graph of the function intersects the horizontal line at more than one point.

Inverse Functions

Since the function is not one-to-one, we cannot find an inverse function. However, we can still find a formula for the inverse of the function.

Finding the Inverse

To find the inverse of the function, we need to solve the equation $y = 3x^2 - 2$ for $x$. This will give us a formula for the inverse function.

Solving for $x$

To solve for $x$, we can start by adding 2 to both sides of the equation:

$$y + 2 = 3x^2$$

Next, we can divide both sides of the equation by 3:

$$\frac{y + 2}{3} = x^2$$

Taking the Square Root

To find the inverse function, we need to take the square root of both sides of the equation:

$$x = \pm \sqrt{\frac{y + 2}{3}}$$

Restricting the Domain

Since the original function is defined for all non-negative real numbers, we need to restrict the domain of the inverse function to ensure that it is one-to-one.

Conclusion

In conclusion, the function $f(x) = 3x^2 - 2, \, x \geq 0$ is not one-to-one. However, we can still find a formula for the inverse function by solving the equation $y = 3x^2 - 2$ for $x$. The inverse function is given by:

$$x = \pm \sqrt{\frac{y + 2}{3}}$$

However, since the original function is defined for all non-negative real numbers, we need to restrict the domain of the inverse function to ensure that it is one-to-one.

Final Answer

The final answer is: $\boxed{No}$

Introduction

In our previous article, we analyzed the function $f(x) = 3x^2 - 2, \, x \geq 0$ and determined whether it is one-to-one. We also found a formula for the inverse function. In this article, we will answer some common questions related to one-to-one functions and inverse functions.

Q&A

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 determine if a function is one-to-one?

A: To determine if a function is one-to-one, you can use the horizontal line test. If a horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one.

Q: What is the horizontal line test?

A: The horizontal line test is a method used to determine if a function is one-to-one. If a horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one.

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

A: To find the inverse of a function, you need to solve the equation $y = f(x)$ for $x$. This will give you a formula for the inverse function.

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

A: A one-to-one function is a function that maps each element of its domain to a unique element in its range. An inverse function is a function that reverses the operation of the original function.

Q: Can a function be both one-to-one and invertible?

A: Yes, a function can be both one-to-one and invertible. In fact, all one-to-one functions are invertible.

Q: How do I know if a function is invertible?

A: To determine if a function is invertible, you need to check if it is one-to-one. If a function is one-to-one, then it is invertible.

Q: What is the relationship between one-to-one functions and inverse functions?

A: One-to-one functions and inverse functions are closely related. In fact, all one-to-one functions are invertible, and all invertible functions are one-to-one.

Q: Can a function be one-to-one but not invertible?

A: No, a function cannot be one-to-one but not invertible. If a function is one-to-one, then it is invertible.

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 need to solve the equation $y = f(x)$ for $x$. This will give you a formula for the inverse function.

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

A: One-to-one functions are significant in mathematics because they are used to define inverse functions. Inverse functions are used to solve equations and to find the solutions to mathematical problems.

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

A: Yes, one-to-one functions can be used in real-world applications. For example, one-to-one functions are used in computer science to define functions that map input data to output data.

Conclusion

In conclusion, one-to-one functions and inverse functions are important concepts in mathematics. One-to-one functions are used to define inverse functions, and inverse functions are used to solve equations and to find the solutions to mathematical problems. We hope that this Q&A article has helped to clarify any questions you may have had about one-to-one functions and inverse functions.