<s>[INST] Find a formula for the inverse of the function. $f(x) = \frac{4x − 1}{2x + 3}$ Please tell me know if my answer is right and whether the steps are correct? Thanks.
$f(x) = \frac{4x − 1}{2x + 3}$
Step 1: Write $y=f(x)$
$y=\frac{4x-1}{2x+3}$
Step 2: Solve this equation for $x$ in terms of $y$ (if possible)
2(a) Multiply both sides by $2x+3$ 
$(2x+3)\cdot(y)\ =\frac{4x-1}{2x+3}\cdot(2x+3)$
2(b) Distribute y term 
$2xy+3y = 4x-1$
2(c) Isolate  $x$ and $y$ terms
$2xy + 3y = 4x - 1 $
$-2xy+1 = -2xy$
$3y+1 = 4x-2xy$
$3y+1 = x (4-2y)$
2(d) Divide both sides by $(4-2y)$
$x=\frac{3y+1}{4-2y}$
$f^{-1}=\frac{3x+1}{4-2x} $
 [/INST] Your process is correct and your result is correct. You could however stand to delete the line $-2xy+1=-2xy$ from the proof. I get that you were demonstrating what you did to both sides, but it doesn't make sense with the equals sign, and it's possible to go directly from $2xy+3y=4x-1$ to $3y+1=4x-2xy$ anyway.
You can always double check you are correct by plugging in a few value of $x$ and $y$. For example, we can try $x=0$ in the first equation, which gives $y=-\frac{1}{3}$, and plugging in $x=-\frac13$ into the $f^{-1}$ equation gives you $0$ as desired.
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