What is meant by implicit/explicit occurrence of a variable?

[arroy_0205]

I have not understood what precisely is meant when someone says for example, x occurs implicitly or explicitly in a given function. My vague idea is that if f(x) can be expressed solely as a function of x alone then x is said to be explicitly appearing. Is that correct? For exaple in,
$$f(x)=x^{\frac{1}{3}}+\tanh^{-1}e^x+\sinh(\frac{1}{x})+x^x+x \log x$$
I think x appears explicitly.
What about
$$x^y+y^x=1$$
I think in this case it is incorrect to say that here x has explicit dependence in y(x). Can anybody make these precise or correct me if I am wrong?

 

[HallsofIvy]

We should be able to write any function (of x) in the form y= f(x). In order to determine whether the y in x^y+ y^x= 1, you have to try to "solve for y". Whether you can or not, y is NOT and "explicit" function of x.