Question

How to solve, in a one-to-one format, meaning, checking if this function is a one-to-one function or not.

f(x) = (x - 1)/(3x + 3)

Now I know what the restrictions on a one-to-one function are, and I understand all it's concepts. But my biggest issue is solving for x1 and x2, to find if they are equal to each other or not. This is really more of an algebra question than anything.

I just want someone to help me solve this function, to find if they are equal to each other or not.

Answer 1

For \(f\) to be one-to-one, it has to satisfy \(f(x_1)=f(x_2)\implies x_1=x_2\).

So given
\[\frac{x_1-1}{3x_1+3}=\frac{x_2-1}{3x_2+3}\]you need to show that this means necessarily that \(x_1=x_2\). Let's multiply both sides by \(3\) to get rid of that factor of \(3\) in the denominator, then do some algebraic rearranging:
\[\begin{align*}
\frac{x_1-1}{x_1+1}&=\frac{x_2-1}{x_2+1}\\[1ex]
(x_1-1)(x_2+1)&=(x_1+1)(x_2-1)\\[1ex]
&~~\vdots
\end{align*}\]See what happens when you expand both sides and simplify.