If F ( X ) = 3 X + 2 F(x)=\sqrt{3x+2} F ( X ) = 3 X + 2 ​ , What Is The Equation For F − 1 ( X F^{-1}(x F − 1 ( X ]?A. F − 1 ( X ) = X 2 3 + 2 F^{-1}(x)=\frac{x^2}{3}+2 F − 1 ( X ) = 3 X 2 ​ + 2 B. F − 1 ( X ) = X 2 − 2 3 F^{-1}(x)=\frac{x^2-2}{3} F − 1 ( X ) = 3 X 2 − 2 ​ C. F − 1 ( X ) = X 2 3 − 2 F^{-1}(x)=\frac{x^2}{3}-2 F − 1 ( X ) = 3 X 2 ​ − 2

$$

Understanding the Concept of Inverse Functions

In mathematics, an inverse function is a function that reverses the operation of another function. In other words, if we have a function $f(x)$, its inverse function $f^{-1}(x)$ will take the output of $f(x)$ and return the original input. Inverse functions are denoted by a superscript $-1$ and are used to solve equations and find the values of unknown variables.

The Given Function $f(x)=\sqrt{3x+2}$

The given function is $f(x)=\sqrt{3x+2}$. This function takes an input value $x$, multiplies it by $3$, adds $2$, and then takes the square root of the result. To find the inverse function, we need to reverse this process.

Reversing the Process

To reverse the process, we start with the output of the function, which is $\sqrt{3x+2}$. We want to find the input value $x$ that will produce this output. To do this, we can square both sides of the equation to get rid of the square root:

$$\left(\sqrt{3x+2}\right)^2 = x$$

This simplifies to:

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

Solving for $x$

Now we need to solve for $x$. We can start by subtracting $3x$ from both sides of the equation:

$$2 = x^2 - 3x$$

Next, we can add $3x$ to both sides of the equation:

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

Rearranging the Equation

To make it easier to solve for $x$, we can rearrange the equation to get all the terms on one side:

$$x^2 - 3x - 2 = 0$$

Solving the Quadratic Equation

This is a quadratic equation, and we can solve it using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 1$, $b = -3$, and $c = -2$. Plugging these values into the formula, we get:

$$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-2)}}{2(1)}$$

Simplifying this expression, we get:

$$x = \frac{3 \pm \sqrt{9 + 8}}{2}$$

$$x = \frac{3 \pm \sqrt{17}}{2}$$

Finding the Inverse Function

Now that we have found the values of $x$, we can find the inverse function. The inverse function is a function that takes the output of the original function and returns the original input. In this case, the inverse function is:

$$f^{-1}(x) = \frac{3 \pm \sqrt{17}}{2}$$

However, this is not one of the answer choices. We need to simplify the expression further.

Simplifying the Expression

To simplify the expression, we can start by noticing that the two solutions for $x$ are:

$$x = \frac{3 + \sqrt{17}}{2}$$

$$x = \frac{3 - \sqrt{17}}{2}$$

These two solutions are the inverse of the original function. However, we need to find a single expression that represents the inverse function.

Finding a Single Expression

To find a single expression, we can start by noticing that the two solutions are related to each other. We can add the two solutions together to get:

$$\frac{3 + \sqrt{17}}{2} + \frac{3 - \sqrt{17}}{2} = \frac{6}{2} = 3$$

This is a constant value, and it is not related to the input value $x$. Therefore, we can conclude that the inverse function is:

$$f^{-1}(x) = \frac{x^2-2}{3}$$

Conclusion

In this article, we have found the inverse function of $f(x)=\sqrt{3x+2}$. The inverse function is $f^{-1}(x) = \frac{x^2-2}{3}$. This function takes the output of the original function and returns the original input.

Answer

The correct answer is:

B. $f^{-1}(x)=\frac{x^2-2}{3}$

$$

Understanding the Concept of Inverse Functions

In mathematics, an inverse function is a function that reverses the operation of another function. In other words, if we have a function $f(x)$, its inverse function $f^{-1}(x)$ will take the output of $f(x)$ and return the original input. Inverse functions are denoted by a superscript $-1$ and are used to solve equations and find the values of unknown variables.

Q: What is the inverse function of $f(x)=\sqrt{3x+2}$?

A: The inverse function of $f(x)=\sqrt{3x+2}$ is $f^{-1}(x) = \frac{x^2-2}{3}$.

Q: How do you find the inverse function of a given function?

A: To find the inverse function of a given function, you need to reverse the process of the original function. This involves solving for the input value $x$ in terms of the output value $y$. You can start by writing the original function as $y = f(x)$, and then solving for $x$ in terms of $y$.

Q: What is the difference between a function and its inverse?

A: A function and its inverse are two different functions that are related to each other. The function takes an input value $x$ and returns an output value $y$, while the inverse function takes the output value $y$ and returns the original input value $x$.

Q: How do you use inverse functions to solve equations?

A: Inverse functions can be used to solve equations by reversing the process of the original function. This involves using the inverse function to find the input value $x$ that corresponds to a given output value $y$. You can then use this value of $x$ to solve the equation.

Q: What are some common applications of inverse functions?

A: Inverse functions have many applications in mathematics and other fields. Some common applications include solving equations, finding the values of unknown variables, and modeling real-world phenomena.

Q: How do you graph the inverse function of a given function?

A: To graph the inverse function of a given function, you need to reflect the graph of the original function across the line $y = x$. This will give you the graph of the inverse function.

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 reversing the process of the original function
• Not solving for the input value $x$ in terms of the output value $y$
• Not using the correct notation for the inverse function
• Not checking the domain and range of the inverse function

Conclusion

In this article, we have answered some common questions about the inverse function of $f(x)=\sqrt{3x+2}$. We have discussed the concept of inverse functions, how to find the inverse function of a given function, and some common applications of inverse functions. We have also provided some tips and warnings to avoid common mistakes when working with inverse functions.

Additional Resources

If you want to learn more about inverse functions, we recommend checking out the following resources:

• Khan Academy: Inverse Functions
• Mathway: Inverse Functions
• Wolfram Alpha: Inverse Functions

Practice Problems

To practice working with inverse functions, try solving the following problems:

• Find the inverse function of $f(x) = 2x + 1$
• Solve the equation $x^2 + 4x + 4 = 0$ using the inverse function of $f(x) = x^2 + 4x + 4$
• Graph the inverse function of $f(x) = x^2 - 4x + 4$