Which Of The Following Functions Is The Inverse Of F ( X ) = 5 X 2 − 3 F(x) = 5x^2 - 3 F ( X ) = 5 X 2 − 3 ?A. F − 1 ( X ) = X 5 + 3 F^{-1}(x) = \sqrt{\frac{x}{5} + 3} F − 1 ( X ) = 5 X ​ + 3 ​ B. F − 1 ( X ) = X + 3 5 F^{-1}(x) = \frac{\sqrt{x + 3}}{5} F − 1 ( X ) = 5 X + 3 ​ ​ C. F − 1 ( X ) = X + 3 5 F^{-1}(x) = \sqrt{\frac{x + 3}{5}} F − 1 ( X ) = 5 X + 3 ​ ​ D. $f^{-1}(x) =

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Introduction

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) that maps an input x to an output f(x), then the inverse function f^(-1)(x) maps the output f(x) back to the input x. In this article, we will explore how to find the inverse of a quadratic function, specifically the function f(x) = 5x^2 - 3.

What is a Quadratic Function?

A quadratic function is a polynomial function of degree two, which means that the highest power of the variable (in this case, x) is two. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. In our case, the quadratic function is f(x) = 5x^2 - 3.

Finding the Inverse of a Quadratic Function

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

  1. Switch x and y: The first step is to switch the x and y variables in the function. This means that we replace x with y and y with x. So, the function f(x) = 5x^2 - 3 becomes x = 5y^2 - 3.
  2. Solve for y: The next step is to solve for y in the equation x = 5y^2 - 3. We can do this by isolating y on one side of the equation.
  3. Take the square root: Once we have isolated y, we need to take the square root of both sides of the equation to find the inverse function.

Step 1: Switch x and y

Let's start by switching x and y in the function f(x) = 5x^2 - 3. This gives us x = 5y^2 - 3.

Step 2: Solve for y

Now, we need to solve for y in the equation x = 5y^2 - 3. We can do this by adding 3 to both sides of the equation, which gives us x + 3 = 5y^2. Next, we can divide both sides of the equation by 5, which gives us (x + 3)/5 = y^2.

Step 3: Take the square root

Finally, we need to take the square root of both sides of the equation (x + 3)/5 = y^2. This gives us y = ±√((x + 3)/5).

Simplifying the Inverse Function

We can simplify the inverse function y = ±√((x + 3)/5) by removing the ± sign. Since the square root function is defined only for non-negative numbers, we can assume that the input x is non-negative. This gives us y = √((x + 3)/5).

Comparing the Options

Now, let's compare the options given in the problem:

A. f^(-1)(x) = √((x + 3)/5) B. f^(-1)(x) = √(x + 3)/5 C. f^(-1)(x) = √((x + 3)/5) D. f^(-1)(x) = √(x + 3)/5

We can see that options A and C are the same, and options B and D are the same. However, option A is the correct answer, since it matches the simplified inverse function we derived.

Conclusion

In conclusion, we have found the inverse of the quadratic function f(x) = 5x^2 - 3. The inverse function is y = √((x + 3)/5). We compared the options given in the problem and found that option A is the correct answer.

Final Answer

The final answer is:

Introduction

In our previous article, we explored how to find the inverse of a quadratic function. In this article, we will answer some common questions about inverse functions and provide additional examples to help you understand the concept.

Q: What is the purpose of finding the inverse of a function?

A: The purpose of finding the inverse of a function is to reverse the operation of the original function. In other words, if we have a function f(x) that maps an input x to an output f(x), then the inverse function f^(-1)(x) maps the output f(x) back to the input x.

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

A: A function has an inverse if it is one-to-one, meaning that each output value corresponds to exactly one input value. In other words, if a function passes the horizontal line test, then it has an inverse.

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

A: A function and its inverse are like two sides of the same coin. The function maps an input x to an output f(x), while the inverse function maps the output f(x) back to the input x.

Q: Can a function have more than one inverse?

A: No, a function cannot have more than one inverse. The inverse of a function is unique, meaning that there is only one way to reverse the operation of the original function.

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

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

  1. Switch x and y: Switch the x and y variables in the function.
  2. Solve for y: Solve for y in the equation.
  3. Take the square root: Take the square root of both sides of the equation.

Q: What if the function is not quadratic?

A: If the function is not quadratic, then you will need to use a different method to find its inverse. For example, if the function is a linear function, then you can simply switch the x and y variables and solve for y.

Q: Can I use a calculator to find the inverse of a function?

A: Yes, you can use a calculator to find the inverse of a function. However, keep in mind that the calculator may not always give you the correct answer, especially if the function is complex.

Q: What are some common mistakes to avoid when finding the inverse of a function?

A: Some common mistakes to avoid when finding the inverse of a function include:

  • Not switching x and y: Make sure to switch the x and y variables in the function.
  • Not solving for y: Make sure to solve for y in the equation.
  • Not taking the square root: Make sure to take the square root of both sides of the equation.

Q: How do I check if my answer is correct?

A: To check if your answer is correct, you can plug in a value for x and see if the output matches the original function. You can also use a calculator to check your answer.

Conclusion

In conclusion, finding the inverse of a function is an important concept in mathematics. By following the steps outlined in this article, you can find the inverse of a function and understand the concept better. Remember to avoid common mistakes and check your answer to ensure that it is correct.

Final Tips

  • Practice, practice, practice: The more you practice finding the inverse of a function, the better you will become.
  • Use a calculator: A calculator can be a useful tool when finding the inverse of a function.
  • Check your answer: Make sure to check your answer to ensure that it is correct.

Common Inverse Functions

Here are some common inverse functions:

  • Inverse of a linear function: If the function is a linear function, then the inverse function is simply the original function.
  • Inverse of a quadratic function: If the function is a quadratic function, then the inverse function is the square root of the original function.
  • Inverse of a polynomial function: If the function is a polynomial function, then the inverse function is the reciprocal of the original function.

Conclusion

In conclusion, finding the inverse of a function is an important concept in mathematics. By following the steps outlined in this article, you can find the inverse of a function and understand the concept better. Remember to avoid common mistakes and check your answer to ensure that it is correct.