Name of this type of function.

[Square1]

Hi all. Can someone tell me the name for the following type of function:

y = a^(1/x)

where:
a = a constant
x = the variable...

Also, am I right, that the next function is called an exponential function?

y = a^x

Are the two functions related? Is the first some type of subset of the second?

Thank-ya!

 

[Gusaroff]

Yes, that's the exponential functions and the second (y = a^x) is common case of the first (y = a^(1/x) ). Therefore they have similar characteristics. In this particular case function number 1 will increase or decrease (it depends on sign of the constant "a") less intensively than function number 2.

 

[Square1]

Ok so both are exponential because the variable x is somewhere with the exponent.

One thing though. You said that the second is a case of the first? That's odd I would think that the first is a case of the second since the first has a little bit more complicated looking exponent. Any comments on that?

 

[Gusaroff]

I don't know why really in English language this type of functions called "exponential", because for instance in Russian there are different names for y(x)=x^e ("экспоненциальная функция" = "exponential function") and y(x)=a^x ("показательная функция" ≈ "index-of-power function")

No, the first is a subcase of the second, but the second is GENERAL case of the first)
There is math term known as 'сombined function' and y1(x) = a^(1/x) is combination of y2(z)=a^z and z(x) = 1/x.
In other words:
y2(z) = a^z (here z is intermediate variable) = a^z(x) = a^(1/x) = y2(z(x)) = y1(x).
It's like russian matryoshka))