Simplify $\frac{ae^{-\frac{a}{x}}+be^{-\frac{b}{x}}}{e^{-\frac{a}{x}}+e^{-\frac{b}{x}}}$

Question

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?

Answer 1

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.

Answer 2

$$ y= \frac{ a {\rm e}^{-a/x} + b {\rm e}^{-b/x} }{{\rm e}^{-a/x}+{\rm e}^{-b/x}} = \frac{ a {\rm e}^{a/x} + b {\rm e}^{b/x} }{{\rm e}^{a/x}+{\rm e}^{b/x}}$$

Now consider $$\begin{cases} a & = x \ln(\alpha) \\ b & = x \ln(\beta) \end{cases} $$

That makes the expression into

$$ y = \frac{x}{\alpha + \beta} ( \beta \ln \alpha + \alpha \ln \beta ) $$

That's all I got :-)