How to get the inverse of this function?

Question

I have the function f(x) = (1+8x) / (3-3x).

I have been stuck on trying to get the inverse of this function by isolating for y. I ended up with:

x = (1+8y) / (3-3y) but I am not quite sure where to go from there.

Answer 1

$$ y = \frac{1+8x}{3-3x}$$
$$(3-3x)y = 1+8x$$ (multiplying each side by $3-3x$)
$$8x + 3xy = 3y -1$$ (multiplying out the brackets, adding $3xy$ to each side and subtracting $1$ from each side)
$$x(8+3y) = 3y - 1$$ (taking out a factor of $x$ from LHS)
$$x = \frac{3y-1}{8+3y}$$ (dividing each side by $8+3y$)

Answer 2

I get $x=\frac{4y-1}{3y+8}$. ${}{}$

Answer 3

Here we start with $$f(x) = \frac{1+8x}{4-3x}$$

So we set $y = f(x)$ and solve for $x$ in terms of $y$:

$$y = \frac{1+8x}{4-3x}$$

$$(4-3x)y = 1+8x$$ $$4y - 3xy = 1+8x$$ $$4y - 3xy - 1 = 8x$$ $$4y - 3xy - 1 - 8x = 0$$ $$-3xy - 1 - 8x = -4y$$ $$-3xy -8x = -4y+1$$ $$-3x(y+8/3) = -4y+1$$ $$-3x = \frac{-4y+1}{y+8/3}$$ $$x = \frac{4y-1}{3y+8}$$

What we have essentially done is find a general equation in terms of $x$ and $y$. Then, we reverse the process. Can you see how the general method works?