when does f(x)=f inverse (f to the-1 power) {is 1 over x valid}?

Answer 1

graphically... f = f^(-1) ... the function f equals its inverse ... if the curve is symmetric with respect to the line y = x.

so for every point (a,b) on the curve, the point (b,a) is also on the curve. moreover, f must still be a function.

there are many curves that follow this property

y = k/x is one set of them, k must not be zero.

y = x is also another function

y = sqrt(r^2-x^2) , for 0 along with
y = (r^n - x^n)^(1/n) ... the general case, 0
y = -x + b ... these lines are also its own inverse.

those are some examples of functions that are their own inverses. i believe there are still more out there...


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Answer 2

y=1/x and x =1/y are identical because in both cases xy=1 which is a hyperbola.

Answer 3

all numbers except for 0. you can never divide by 0