Question: I need to show that $(a,b)$ and $(c,\infty)$ have the same cardinality, yet I am not sure what to do. The only example I have to go by that is similar is $(0,1)$ and $(1,\infty)$... And that one used the function $f(x)=1/x$.

Answer: 
You have done the hard part. Can you find a bijecton between $(a,b)$ and $(0,1)$? There is a natural one. Then can you find one between $(1,\infty)$ and $(c,\infty)?$. Then you can just compose the bijections $(a,b)\leftrightarrow (0,1) \leftrightarrow (1,\infty) \leftrightarrow (c,\infty)$. Normally you wouldn't be asked to do it explicitly, you have shown that you could and that is enough.
You could try to transform the interval $(a,b)$ to the interval $(0,1)$ like so:

$(a,b) - a \rightarrow (0,b-a)$

$(0,b-a) \* \frac{1}{b-a} \rightarrow (0,1)$

Then you can use your existing argument for the cardinality of $(0,1)$.