$f(x)=\frac{3x+5}{-6x+2}$ , largest possible domain
Find $f^{-1}(x)$ of this 1-1 function and the domain.
So I wrote the equation as
$$y=\frac{3x+5}{-6x+2}$$
Interchanged x and y, and made y the subject (usual method to find the inverse function) and I got :
$$f^{-1}(x)=\frac{2x-5}{6x+3}$$
The problem I have is finding the domain of this. The only value that this function breaks apart is when
$$6x+3=0 \implies x=-\frac{1}{2}$$
So therefore I think the domain is :
$$\left(-\infty,-\frac{1}{2}\right)\bigcup \left(-\frac{1}{2},\infty\right)$$
But this answer is wrong. Please help me.
The domain of $f$ is $\left(-\infty,\frac{1}{3}\right)\bigcup \left(\frac{1}{3},\infty\right)$ because of the denominator's restriction. Likewise, the range of $f$ is $\left(-\infty,\frac{-1}{2}\right)\bigcup \left(\frac{-1}{2},\infty\right)$ because both numerator and denominator are degree 1 which means horizontal asymptotes occur at $y=\frac{3}{-6}=-\frac1{2}$. Since it is a 1-1 function, the domain of $f$ is the range of $f^{-1}$ and the range of $f$ is the domain of $f^{-1}$. I don't see the issue and believe that you are correct.