Determine The Inverse Function Of $f(x) = 3x^2 - 6, \, X \geq 0$.Enter The Correct Expression In The Box.

Introduction

In mathematics, the concept of inverse functions is crucial in understanding the relationship between two functions. Given a function $f(x)$, the 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 the inverse function $f^{-1}(x)$ maps the output $y$ back to the input $x$. In this article, we will determine the inverse function of the quadratic function $f(x) = 3x^2 - 6, \, x \geq 0$.

Understanding the Quadratic Function

The given quadratic function is $f(x) = 3x^2 - 6, \, x \geq 0$. This function is a quadratic function because it is in the form of $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this case, $a = 3$, $b = 0$, and $c = -6$. The function is defined for $x \geq 0$, which means that the domain of the function is the set of all non-negative real numbers.

Graph of the Quadratic Function

To understand the behavior of the quadratic function, let's graph it. The graph of the function is a parabola that opens upwards because the coefficient of $x^2$ is positive. The vertex of the parabola is at the point $(0, -6)$, which is the minimum point of the function. The graph of the function is symmetric about the y-axis because the coefficient of $x$ is zero.

Finding the Inverse Function

To find the inverse function of $f(x) = 3x^2 - 6, \, x \geq 0$, we need to follow these steps:

1. Replace $f(x)$ with $y$: Replace $f(x)$ with $y$ to get $y = 3x^2 - 6$.
2. Interchange $x$ and $y$: Interchange $x$ and $y$ to get $x = 3y^2 - 6$.
3. Solve for $y$: Solve for $y$ to get $y = \pm \sqrt{\frac{x+6}{3}}$.
4. Restrict the domain: Restrict the domain of the inverse function to $x \geq 0$.

Solving for $y$

To solve for $y$, we need to isolate $y$ in the equation $x = 3y^2 - 6$. We can do this by adding $6$ to both sides of the equation to get $x + 6 = 3y^2$. Then, we can divide both sides of the equation by $3$ to get $\frac{x+6}{3} = y^2$. Finally, we can take the square root of both sides of the equation to get $y = \pm \sqrt{\frac{x+6}{3}}$.

Restricting the Domain

The domain of the inverse function is restricted to $x \geq 0$ because the original function is defined for $x \geq 0$. This means that the inverse function is only defined for $x \geq 0$.

Conclusion

In conclusion, the inverse function of $f(x) = 3x^2 - 6, \, x \geq 0$ is $f^{-1}(x) = \sqrt{\frac{x+6}{3}}$. This function is only defined for $x \geq 0$ because the original function is defined for $x \geq 0$.

Final Answer

Introduction

In our previous article, we determined the inverse function of the quadratic function $f(x) = 3x^2 - 6, \, x \geq 0$. In this article, we will answer some frequently asked questions about the inverse function.

Q: What is the inverse function of a quadratic function?

A: The inverse function of a quadratic function is a function that undoes the action of the quadratic function. In other words, if the quadratic function maps an input $x$ to an output $y$, then the inverse function maps the output $y$ back to the input $x$.

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

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

1. Replace $f(x)$ with $y$: Replace $f(x)$ with $y$ to get $y = ax^2 + bx + c$.
2. Interchange $x$ and $y$: Interchange $x$ and $y$ to get $x = ay^2 + by + c$.
3. Solve for $y$: Solve for $y$ to get $y = \pm \sqrt{\frac{x-c}{a}}$.
4. Restrict the domain: Restrict the domain of the inverse function to the domain of the original function.

Q: What is the domain of the inverse function of a quadratic function?

A: The domain of the inverse function of a quadratic function is the set of all real numbers that are greater than or equal to the minimum value of the original function.

Q: What is the range of the inverse function of a quadratic function?

A: The range of the inverse function of a quadratic function is the set of all real numbers that are greater than or equal to the minimum value of the original function.

Q: Can the inverse function of a quadratic function be a linear function?

A: No, the inverse function of a quadratic function cannot be a linear function. The inverse function of a quadratic function is always a quadratic function.

Q: Can the inverse function of a quadratic function be a rational function?

A: Yes, the inverse function of a quadratic function can be a rational function. For example, if the original function is $f(x) = \frac{x^2}{x+1}$, then the inverse function is $f^{-1}(x) = \frac{x-1}{x}$.

Q: Can the inverse function of a quadratic function be a trigonometric function?

A: No, the inverse function of a quadratic function cannot be a trigonometric function. The inverse function of a quadratic function is always a function that is defined in terms of the original function.

Q: Can the inverse function of a quadratic function be a logarithmic function?

A: No, the inverse function of a quadratic function cannot be a logarithmic function. The inverse function of a quadratic function is always a function that is defined in terms of the original function.

Conclusion

In conclusion, the inverse function of a quadratic function is a function that undoes the action of the quadratic function. The domain and range of the inverse function are the set of all real numbers that are greater than or equal to the minimum value of the original function. The inverse function of a quadratic function can be a quadratic function, but it cannot be a linear function, a rational function, a trigonometric function, or a logarithmic function.

Final Answer

The final answer is: $\boxed{\sqrt{\frac{x+6}{3}}}$