Name a point that lies on the inverse of f(x) = 2^x

Answer 1

The easy way:

Let's find a point on the curve for f(x). Let x = 1. Then, f(x) = 2^1 = 2. So, (1,2) is a point on f.

Now, the inverse function is symmetric about the line y = x. So, all we need to do is swap the x and y coordinates to find a point on f^-1. So, (2,1) is one such point.

The long way:

To find the inverse function, we interchange x and y: x = 2^y. Taking logs (base 2) of both sides gives

log(x) = log(2^y) = y*log(2). So, y = log(x)/log(2), or f^(-1)(x) = log(x)/log(2) is the inverse function.

Now, letting x = 2: f^(-1)(2) = log(2)/log(2) = 1, again giving the point (2,1).

So, what if I asked you to plot the inverse of the function f(x) = x^3 + x/(x^2+1)? You may not be able to calculate the inverse by hand, but you can plot the function f(x) and then by switching the coordinates of each point, obtain the plot for the inverse.