My Favourite Function

People have their favourite colours, football teams (go dogs !) and beaches. Why not your favourite function ?

For me there are many attractive candidates for "my favourite function". For example, I enjoy the simplicity of mx + c, the up and down of x², the surprising plateauing of x³, the lovely endless symmetry of the sinx and cosx and even the quirkiness of complex polynominals (x⁴ + x³ + x² + x). One of my associates is very fond of the hyperbolic functions but personally find their curviness rather artificial (they are just a compound of two other functions). I must admitt that I find the limited domain of most inverse functions a little off putting. Why choose a function with a limited range when you can have the whole number line !

I think looking for a favourite in any area involves the formation of various vanities and snobberies, which is what makes competitions like "the top ten albums of all time" alot of fun. It is an opportunity to laugh at your own prejudices whilst studying the quirky choices of others.

So what is my favourite ?

e^x is definitely my favourite function. Why ?

Certainly, the exponential function forms a pleasing curve but it is more its amazing characteristics that draws me to e^x. I love the fact the function is based on an irrational number but calculates something commonly observed in nature (e.g. radioactive decay, rates of chemical reactions, etc.). I find the idea that the slope of any point of the line is the value at that point (dy/dx = e^x) amazing and totally fascinating. For me, e^x is number one ! (which is only true when x = 0)

What is your favourite function ?