What is the difference between function of x and function of 1/x

Question

Difference between function of x and function of 1/x. 

F(x) and F(1/x)

Answer 1

F(1/x) just means that you plug in 1/x instead of x in the definition of F(x). (This is called composition of functions. 1/x is itself a function of x and you substitute it into F(x). You are likely to see this idea again in the future.)

 

For example, if f(x) = x² + 2x - 1. Then f(1/x) = (1/x)² + 2(1/x) - 1 = 1/x² + 2/x - 1 = (1+2x-x²)/x². There isn't too much you can say generally about the relationship between these two functions. In general, they will behave very differently.

Using calculus, we can say the following by the chain rule: the derivative of F(1/x) is F'(1/x)*(-1/x^2).

Answer 2

Suppose f(x) = x² + 3 (as an example for discussion purposes).

 

 

The first is a defining rule; the second is a substitution, into that rule, putting 1/x in place of x.

 

I do NOT agree with Michael J's statement, in general.

 

 

Answer 3

F(x) and F(1/x) are inversely proportional to each other.