How to Find the Inverse Function of a Linear Equation

Oct 13, 2025 by miodeya.com 122 views

**Finding the Inverse Function of a Linear Equation** =====================================================

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 function of a linear equation.

What is a Linear Equation?

A linear equation is an equation in which the highest power of the variable (usually x) is 1. In other words, a linear equation is an equation of the form ax + b = c, where a, b, and c are constants. For example, the equation 2x + 3 = 5 is a linear equation.

How to Find the Inverse Function of a Linear Equation

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

Switch the x and y variables: The first step in finding the inverse function is to switch the x and y variables. In other words, we replace x with y and y with x.
Solve for y: Once we have switched the x and y variables, we need to solve for y. This involves isolating y on one side of the equation.
Replace y with f^(-1)(x): Finally, we replace y with f^(-1)(x) to get the inverse function.

Example: Finding the Inverse Function of f(x) = -5x - 4

Let's use the function f(x) = -5x - 4 as an example. To find the inverse function, we follow the steps above:

Switch the x and y variables: We replace x with y and y with x to get y = -5x - 4.
Solve for y: To solve for y, we add 4 to both sides of the equation to get y + 4 = -5x.
Replace y with f^(-1)(x): Finally, we replace y with f^(-1)(x) to get f^(-1)(x) = -5x - 4 + 4, which simplifies to f^(-1)(x) = -5x.

However, we need to express the inverse function in terms of x. To do this, we can divide both sides of the equation by -5 to get f^(-1)(x) = -x.

Which Function is the Inverse of f(x) = -5x - 4?

Now that we have found the inverse function of f(x) = -5x - 4, we can compare it to the options given in the problem:

A. f^(-1)(x) = -\frac{1}{5}x - \frac{4}{5} B. f^(-1)(x) = -\frac{1}{5}x + \frac{4}{5} C. f^(-1)(x) = -4x + 5 D. f^(-1)(x) = 4x + 4

The correct answer is B. f^(-1)(x) = -\frac{1}{5}x + \frac{4}{5}.

Conclusion

In this article, we have explored how to find the inverse function of a linear equation. We have used the function f(x) = -5x - 4 as an example and have found its inverse function to be f^(-1)(x) = -\frac{1}{5}x + \frac{4}{5}. We have also compared this result to the options given in the problem and have determined that the correct answer is B.

Frequently Asked Questions

Q: What is the inverse function of a linear equation?

A: The inverse function of a linear equation is a function that reverses the operation of the original function.

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

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

Switch the x and y variables.
Solve for y.
Replace y with f^(-1)(x).

Q: What is the inverse function of f(x) = -5x - 4?

A: The inverse function of f(x) = -5x - 4 is f^(-1)(x) = -\frac{1}{5}x + \frac{4}{5}.

Q: Which function is the inverse of f(x) = -5x - 4?

A: The correct answer is B. f^(-1)(x) = -\frac{1}{5}x + \frac{4}{5}.