What is the Inverse Function of fx (2-3x)/(x 1)?

To find the inverse function of f(x) frac{2 - 3x}{x 1}, we'll follow a systematic approach. Let's break down the steps to find the inverse:

Step 1: Replace fx with y

We start by substituting y frac{2 - 3x}{x 1}.

Step 2: Swap x and y

Next, we swap x and y, resulting in x frac{2 - 3y}{y 1}.

Step 3: Solve for y

To solve for y, we follow these algebraic steps:

Multiply both sides by y 1 to clear the denominator:xy x 2 - 3y. Distribute x and rearrange the equation to isolate terms involving y:xy 3y 2 - x. Factor out y on the left side:y(x 3) 2 - x. Divide both sides by x 3 to isolate y and obtain the inverse function:y frac{2 - x}{x 3}.

The inverse function is f^-1(x) frac{2 - x}{x 3}

Visualization of the Inverse Function

For a graphical representation, observe that each of the functions in red and blue (dependent and inverse) are reflections about the diagonal line y x. Each is the inverse of the other.

Verification of the Inverse Function

Before we begin, it's crucial to verify that the given function has an inverse. We check its domain and range. The domain of the function is all x values except -1. The range of the function is all y values except -3. This means the horizontal asymptote is given by limx to pm infty f(x) -3.

Existence of the inverse is ensured by the bijectivity of the function. The function is bijective over its domain, thus guaranteeing the existence of the inverse.

Conclusion

The inverse function of f(x) is f^-1(x) frac{2 - x}{x 3}