Understanding the Inverse of the Function f(x) (2 - x) / (6 - x)

Understanding the Inverse of the Function f(x) (2 - x) / (6 - x)

Introduction

In mathematics, particularly in algebra, finding the inverse of a function is a fundamental concept. We are discussing the inverse function for the given function f(x) (2 - x) / (6 - x). This article will guide you through the process of finding and understanding the inverse function, and provide necessary algebraic manipulations involved.

The Function and Its Inverse

Given the function:

$$f(x) dfrac{2 - x}{6 - x}$$

Step 1: Expressing y f(x)

First, we express the function in terms of y:

$$y dfrac{2 - x}{6 - x}$$

Step 2: Swapping x and y

Next, we swap x and y to find the inverse function:

$$x dfrac{2 - y}{6 - y}$$

Step 3: Solving for y

To find the inverse function, we need to solve this equation for y. Let's proceed with step-by-step algebraic manipulation:

Click to see detailed steps and algebraic manipulations

Start with the equation:

$$x dfrac{2 - y}{6 - y}$$

Multiply both sides by (6 - y) to eliminate the denominator:

$$x(6 - y) 2 - y$$

Expand and rearrange the equation:

$$6x - xy 2 - y$$

Move all terms involving y to one side of the equation:

$$6x - 2 xy - y$$

Factor out y on the right side:

$$6x - 2 y(x - 1)$$

Finally, solve for y:

$$y dfrac{6x - 2}{x - 1}$$

Thus, the inverse function is:

$$f^{-1}(x) dfrac{6x - 2}{x - 1}$$

Verification of the Inverse Function

To verify that we have found the correct inverse function, we can substitute (f(f^{-1}(x))) back into the original function and check if we get (x).

$$fleft(dfrac{6x - 2}{x - 1}right) dfrac{2 - left(dfrac{6x - 2}{x - 1}right)}{6 - left(dfrac{6x - 2}{x - 1}right)}$$

Step 1: Simplify the numerator

$$2 - dfrac{6x - 2}{x - 1} dfrac{2(x - 1) - (6x - 2)}{x - 1} dfrac{2x - 2 - 6x 2}{x - 1} dfrac{-4x}{x - 1}$$

Step 2: Simplify the denominator

$$6 - dfrac{6x - 2}{x - 1} dfrac{6(x - 1) - (6x - 2)}{x - 1} dfrac{6x - 6 - 6x 2}{x - 1} dfrac{-4}{x - 1}$$

Step 3: Combine the simplified numerator and denominator

$$dfrac{dfrac{-4x}{x - 1}}{dfrac{-4}{x - 1}} dfrac{-4x}{-4} x$$

This verifies that (f(f^{-1}(x)) x), confirming that our inverse function is correct.

Conclusion

We have successfully found and verified the inverse of the function f(x) (2 - x) / (6 - x). The inverse function is: