Inverse Function: f(x) = (3+4x)/4x

Question

Find the inverse of the function:

$$\ f(x)= \frac{3+4x}{x−1}$$

My answer:

$$\ = \frac{3}{4x−4}$$

I have been informed that my answer is incorrect.

Can anyone explain this?

Answer 1

Finding the inverse function, let's start off of the basic expression : $$y=f(x) \Leftrightarrow f^{-1}(y) = x \Rightarrow y = f^{-1}(x)$$

So :

$$y = \frac{3+4x}{x-1} \Leftrightarrow y(x-1) = 3+4x \Leftrightarrow yx - y - 3 - 4x = 0 \Leftrightarrow(y-4)x-y-3=0 $$

$$\Leftrightarrow$$

$$(y-4)x = y+3 \Leftrightarrow x = \frac{y+3}{y-4}$$

which stands for $y-4 \neq 0$ but also take into account that $x \neq 1$ from the initial domain of $f$.

Thus, the inverse is :

$$g(x) = f^{-1}(x) = \frac{x+3}{x-4}, \space x\neq4, x\neq 1$$

Answer 2

$$\frac{3+4x}{x-1}=y\implies xy-y-4x=3\implies x(y-4)=y+3\implies$$

$$x=\frac{y+3}{y-4}$$

and thus the inverse function is

$$g(x)=\frac{x+3}{x-4}$$

Check it.