Solving equations of type
First, I'm a new person on this site, so please correct me if I'm asking the question in a wrong way.
I thought I'm not a big fan of maths, but recently I've stumbled upon one interesting fact, which I'm trying to find an explanation for. I've noticed that graphs of functions and , where is given and equal for both functions, always have intersection points. This means, equation must have solutions, at least it's what I see from the graphs.
I've tried to solve this equation analytically for some given , like , but my skills are very rusty, and I cannot come up with anything. So I'm here for help, and my question(-s) are:
are these functions always have intersection points?
if yes, why, if not, when not?
how to solve equations like analytically?