How to Find the Inverse of the Function f(x) 2x/3 - 6

Understanding Inverse Functions: Inverse functions are critical in mathematics, particularly in algebra, as they help us to reverse operations or find the original value given the result.

1. The Original Function and Its Role

Consider the function f(x) 2x/3 - 6. This function can be described as a transformation that takes an input x and produces an output. The goal is to find a function that would reverse the operations performed by f(x).

2. Steps to Find the Inverse Function

Step 1: Replace f(x) with y

First, we start by replacing f(x) with y:

y 2x/3 - 6

Step 2: Swap x and y

Next, we swap y and x:

x 2y/3 - 6

Step 3: Solve for y

Now, we solve for y:

Add 6 to both sides of the equation: Multiply both sides by 3 to eliminate the fraction: Divide both sides by 2: Simplify the expression:

Step 3.1: Add 6 to both sides:

x 6 2y/3

Step 3.2: Multiply both sides by 3 to eliminate the fraction:

3(x 6) 2y

Step 3.3: Divide both sides by 2:

y 3/2(x 6)

Step 3.4: Simplify the expression:

y 3x/2 18/2

y 3x/2 9

Step 4: Replace y with f-1(x)

Finally, we replace y with f-1(x):

f-1(x) 3x/2 9

Summary of the Inverse Function

The inverse function of f(x) 2x/3 - 6 is:

Conclusion

By following these steps, we can systematically find the inverse of a given function. This process is crucial in various mathematical applications, including solving equations and modeling real-world scenarios within the field of algebra.

Additional Resources and Further Reading

For more in-depth learning and practice, consider studying more about functions and their inverses, and exploring related topics in algebra.