How to Determine the Inverse Function of g^-1(x) 4 / 5x^2

In this article, we will explore the concept of inverse functions, specifically how to determine the inverse function for a given mathematical function. We will use the function g^-1(x) 4 / 5x^2 to illustrate the process. This exploration will cover the steps required to convert this function into its inverse form and provide a practical example of the algebraic manipulation involved.

Understanding Inverse Functions

An inverse function is a function that "reverses" another function. If you have a function y f(x), its inverse function g(x) is such that g(f(x)) x for every x in the domain of f, and f(g(x)) x for every x in the domain of g. In simpler terms, applying a function and its inverse successively should return the original input.

Given Function and Its Inverse

The given function is:

g^-1(x) 4 / 5x^2

This function is given in a form that suggests it is the inverse of some function g(x). We are also told that if we substitute 4/5x^2 for y, we get:

x 4 - 2y / 5y

Let's break down how we arrived at this and then use it to find the original function g(x).

Step-by-Step Solution

1. Start with the given equation:

g^-1(x) 4 / 5x^2

2. Let 4/5x^2 y. Therefore, we have:

y 4 / 5x^2

3. Solve for x in terms of y using the given equation:

x 4 - 2y / 5y

4. Simplify the right-hand side of the equation:

x 4 - (2y / 5y)

5. Further simplify the fraction:

x 4 - 2 / 5

6. Combine the terms:

x (20 - 2) / 5

x 18 / 5

While this approach does not give a clear inverse function for our original variable y, we can see that the inverse function of g^-1(x) 4 / 5x^2 is not directly derivable in a simple form. Instead, we can use the given relationship to express g(y) in terms of y.

Expressing g(y) in Terms of y

Given that:

x 4 - 2y / 5y

We can rewrite this in terms of g(y) as follows:

g(y) 4 - 2y / 5y

This form of g(y) is not the inverse function in the strictest sense, but it represents the function in terms of its output.

Conclusion

In conclusion, the given function g^-1(x) 4 / 5x^2 is not inverted in a straightforward manner to directly produce an inverse function. Instead, we have demonstrated the process of algebraic manipulation to express the function in terms of its output variable, providing insight into how to approach such problems.

This exploration highlights the importance of understanding the relationship between a function and its inverse, and the need for careful algebraic steps in determining these relationships. For more complex functions, these steps can be extended to find more generalized solutions.

Understanding these concepts and practicing similar problems can significantly enhance one's skills in mathematical analysis and problem-solving.