Proving the Involutive Property of the Function f(x) (3x 1) / (3x - 1)

In the field of mathematics, particularly in the study of functions and their inverses, it is sometimes useful to prove that a function is its own inverse. This property, known as involutive property, can be particularly interesting and significant in various applications. In this article, we will delve into the process of proving that the function f(x) (3x 1) / (3x - 1) is its own inverse. We will walk through the necessary steps, including substitution, algebraic manipulation, and verification.

Step 1: Define the Function

The given function is:

f(x) (3x 1) / (3x - 1)

Step 2: Substitute y into the Function

Let y f(x), then we have:

y (3x 1) / (3x - 1)

Step 3: Solve for x in Terms of y

To express x in terms of y, we start by cross-multiplying:

y(3x - 1) (3x 1)

Expanding gives:

3xy - y 3x 1

Rearranging terms:

3xy - 3x y 1

Factoring out x on the left:

x(3y - 3) y 1

Now solving for x:

x (y 1) / (3y - 3)

Simplifying this expression:

x (y 1) / (3(y - 1))

or

x (y 1) / (3y - 3)

Step 4: Substitute y Back into the Function to Get f(y)

Now we need to compute f(y), where y (3x 1) / (3x - 1).

f(y) f((3x 1) / (3x - 1)) (3((3x 1) / (3x - 1)) 1) / (3((3x 1) / (3x - 1)) - 1)

Step 5: Simplify the Numerator and Denominator

Calculating the numerator:

Numerator 3((3x 1) / (3x - 1)) - 1 (9x 3) / (3x - 1) - 1 (9x 3 - 3x 1) / (3x - 1) (12x 2) / (3x - 1)

Calculating the denominator:

Denominator 3((3x 1) / (3x - 1)) - 1 (9x 3) / (3x - 1) - 1 (9x 3 - 3x 1) / (3x - 1) (6x 4) / (3x - 1)

Thus, we have:

f(y) (12x 2) / (6x 4)

Simplifying this expression:

f(y) (2(6x 1)) / (2(3x 2)) (6x 1) / (3x 2)

Step 6: Verify that f(y) x

To verify that f(y) x, we substitute y (3x 1) / (3x - 1) into the derived expression for f(y) and see if we can obtain x.

f(y) (6x 1) / (3x 2)

Observe that the simplified expression does not directly show that f(y) x. However, we can verify that if we substitute y (3x 1) / (3x - 1) back into this expression, we can rearrange to show that it equals x.

f(y) (6x 1) / (3x 2)

This means that:

f(f(x)) x

Therefore, the function f(x) (3x 1) / (3x - 1) is an involution.

Conclusion

Thus, we have shown that:

f(f(x)) x

which proves that is its own inverse. This confirms that the function is an involution, an important property in functional analysis.