Simplify $\frac{ae^{-\frac{a}{x}}+be^{-\frac{b}{x}}}{e^{-\frac{a}{x}}+e^{-\frac{b}{x}}}$ Do you guys have any way of simplifying: $$f(x) =\frac{ae^{-\frac{a}{x}}+be^{-\frac{b}{x}}}{e^{-\frac{a}{x}}+e^{-\frac{b}{x}}}?$$ I am having a hard time fining a way to visualize this function. Is there any way I could change the constants to make it a cosh?

Since the exponent are different it seems not convenient to simplify $f(x)$ by $cosh x$, instead we can obtain

$$f(x) =\frac{ae^{-\frac{a}{x}}+be^{-\frac{b}{x}}}{e^{-\frac{a}{x}}+e^{-\frac{b}{x}}} =\frac{ae^{-\frac{a}{x}}+ae^{-\frac{b}{x}}+(b-a)e^{-\frac{b}{x}}}{e^{-\frac{a}{x}}+e^{-\frac{b}{x}}}=a+(b-a)\frac{e^{-\frac{b}{x}}}{e^{-\frac{a}{x}}+e^{-\frac{b}{x}}}=\\\=a+(b-a)\frac{1}{e^{-\frac{a-b}{x}}+1}$$

which is not so bad since we have only an exponential term and all others constant.