Inverse Function Drawing

When we graph the inverse function of some function f(x), we see that it is actually the mirror of f along the line x=y. It makes sense! From the definition of inverse functions, we know that for each (a,b) in f(x) there will be a (b,a) in f^-1(x). For example, look at the following:

In this example, the blue function is x² and the red function is the square root of x. We know how x² looks like - how will we graph its inverse? We will take some points we know on the first graph (for example, the points (0,0), (1/10,1/100), (1/2,1/4), (1,1), (3/2,9/4) and so on), and for each point we will draw a red point which has the opposite coordinates. For example, if we had (1/2,1/4) we will draw a red point in (1/4,1/2). Then we connect all of the red points, and get the wanted red graph!

Another option is to think we are the mirror of the original graph, always. This is how it works fast: (try to understand!)

The Logarithm

We also know that the logarithm is the inverse function of the exponent. This is how it looks like for 2^x and log₂x:

Exercise

In order to practice that, take an invertible function you know (doesn't matter exactly which), and invert it in the following way:

• Make a list of points of the form (x,f(x))
• Draw them as blue points
• Make an analogous list of points of the form (f(x),x)
• Draw them as red points
• Connect the blue points to create a blue graph and the red points to create a red graph. The red graph should approximately be the inverse graph of the blue one!

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